https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Pmwood&feedformat=atomUW-Math Wiki - User contributions [en]2020-02-21T16:51:36ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17464Probability Seminar2019-05-23T14:16:32Z<p>Pmwood: /* September 12, 2019, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 5, 2019, TBA ==<br />
== September 12, 2019, TBA ==<br />
<br />
== September 19, 2019, TBA ==<br />
<br />
<!-- == September 26, 2019, TBA == --><br />
<br />
== October 3, 2019, TBA ==<br />
<br />
== October 10, 2019, TBA ==<br />
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== October 17, 2019, TBA ==<br />
<br />
== October 24, 2019, TBA ==<br />
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== October 31, 2019, TBA ==<br />
<br />
== November 7, 2019, TBA ==<br />
<br />
== November 14, 2019, TBA ==<br />
<br />
== November 21, 2019, TBA ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
== December 5, 2019, TBA ==<br />
<br />
<br />
<br />
<!--<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17463Probability Seminar2019-05-23T14:13:16Z<p>Pmwood: /* December 19, 2019, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, TBA ==<br />
<br />
== September 19, 2019, TBA ==<br />
<br />
<!-- == September 26, 2019, TBA == --><br />
<br />
== October 3, 2019, TBA ==<br />
<br />
== October 10, 2019, TBA ==<br />
<br />
== October 17, 2019, TBA ==<br />
<br />
== October 24, 2019, TBA ==<br />
<br />
== October 31, 2019, TBA ==<br />
<br />
== November 7, 2019, TBA ==<br />
<br />
== November 14, 2019, TBA ==<br />
<br />
== November 21, 2019, TBA ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
== December 5, 2019, TBA ==<br />
<br />
<br />
<br />
<!--<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17462Probability Seminar2019-05-23T14:13:07Z<p>Pmwood: /* December 12, 2019, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, TBA ==<br />
<br />
== September 19, 2019, TBA ==<br />
<br />
<!-- == September 26, 2019, TBA == --><br />
<br />
== October 3, 2019, TBA ==<br />
<br />
== October 10, 2019, TBA ==<br />
<br />
== October 17, 2019, TBA ==<br />
<br />
== October 24, 2019, TBA ==<br />
<br />
== October 31, 2019, TBA ==<br />
<br />
== November 7, 2019, TBA ==<br />
<br />
== November 14, 2019, TBA ==<br />
<br />
== November 21, 2019, TBA ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
== December 5, 2019, TBA ==<br />
<br />
== December 19, 2019, TBA ==<br />
<br />
<br />
<br />
<!--<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
--><br />
<br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17461Probability Seminar2019-05-23T14:12:52Z<p>Pmwood: /* September 26, 2019, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, TBA ==<br />
<br />
== September 19, 2019, TBA ==<br />
<br />
<!-- == September 26, 2019, TBA == --><br />
<br />
== October 3, 2019, TBA ==<br />
<br />
== October 10, 2019, TBA ==<br />
<br />
== October 17, 2019, TBA ==<br />
<br />
== October 24, 2019, TBA ==<br />
<br />
== October 31, 2019, TBA ==<br />
<br />
== November 7, 2019, TBA ==<br />
<br />
== November 14, 2019, TBA ==<br />
<br />
== November 21, 2019, TBA ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
== December 5, 2019, TBA ==<br />
<br />
== December 12, 2019, TBA ==<br />
<br />
== December 19, 2019, TBA ==<br />
<br />
<br />
<br />
<!--<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
--><br />
<br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17460Probability Seminar2019-05-23T14:01:06Z<p>Pmwood: </p>
<hr />
<div>__NOTOC__<br />
<br />
= Fall 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
== September 12, 2019, TBA ==<br />
<br />
== September 19, 2019, TBA ==<br />
<br />
== September 26, 2019, TBA ==<br />
<br />
== October 3, 2019, TBA ==<br />
<br />
== October 10, 2019, TBA ==<br />
<br />
== October 17, 2019, TBA ==<br />
<br />
== October 24, 2019, TBA ==<br />
<br />
== October 31, 2019, TBA ==<br />
<br />
== November 7, 2019, TBA ==<br />
<br />
== November 14, 2019, TBA ==<br />
<br />
== November 21, 2019, TBA ==<br />
<br />
== November 28, 2019, Thanksgiving (no seminar) ==<br />
<br />
== December 5, 2019, TBA ==<br />
<br />
== December 12, 2019, TBA ==<br />
<br />
== December 19, 2019, TBA ==<br />
<br />
<br />
<br />
<!--<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
--><br />
<br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2019&diff=17459Past Probability Seminars Spring 20192019-05-23T13:50:54Z<p>Pmwood: /* Past Seminars Spring 2019 */</p>
<hr />
<div>[[Probability Seminar | Back to Current Probability Seminar Schedule ]]<br />
<br />
<br />
[[Past Seminars | Back to Past Seminars]]<br />
<br />
<br />
= Past Seminars Spring 2019 =<br />
<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Probability Seminar | Back to Current Probability Seminar Schedule ]]<br />
<br />
<br />
[[Past Seminars | Back to Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2019&diff=17458Past Probability Seminars Spring 20192019-05-23T13:50:38Z<p>Pmwood: /* Past Seminars Spring 2019 */</p>
<hr />
<div>= Past Seminars Spring 2019 =<br />
<br />
[[Probability Seminar | Back to Current Probability Seminar Schedule ]]<br />
<br />
<br />
[[Past Seminars | Back to Past Seminars]]<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Probability Seminar | Back to Current Probability Seminar Schedule ]]<br />
<br />
<br />
[[Past Seminars | Back to Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Past_Probability_Seminars_Spring_2019&diff=17457Past Probability Seminars Spring 20192019-05-23T13:50:09Z<p>Pmwood: Created page with "= Past Seminars Spring 2019 = Back to Current Probability Seminar Schedule Back to Past Seminars <b>Thursdays in 901 Van Vleck..."</p>
<hr />
<div>= Past Seminars Spring 2019 =<br />
<br />
[[Probability Seminar | Back to Current Probability Seminar Schedule ]]<br />
[[Past Seminars | Back to Past Seminars]]<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Probability Seminar | Back to Current Probability Seminar Schedule ]]<br />
[[Past Seminars | Back to Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Past_Seminars&diff=17456Past Seminars2019-05-23T13:46:32Z<p>Pmwood: </p>
<hr />
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<!-- [http://www.math.wisc.edu/~probsem/list-old-sem.html Webpage for older past probability seminars] --></div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17455Probability Seminar2019-05-23T13:45:17Z<p>Pmwood: /* Tuesday , May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:250px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
Title: '''The directed landscape'''<br />
<br />
Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag.<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17411Probability Seminar2019-04-30T19:52:27Z<p>Pmwood: /* Tuesday , May 7, Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:220px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17410Probability Seminar2019-04-30T19:52:19Z<p>Pmwood: /* May 7, Tuesday Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:220px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17409Probability Seminar2019-04-30T19:51:49Z<p>Pmwood: /* May 7, Tuesday Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 7, <span style="color:red">'''Tuesday''' </span> Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:220px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17408Probability Seminar2019-04-30T19:51:35Z<p>Pmwood: /* May 7, Tuesday Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 7, <span style="color:red">'''Tuesday''' </span> Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day. <br />
&emsp; </span></b><br />
</div><br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17407Probability Seminar2019-04-30T19:50:38Z<p>Pmwood: /* May 7, Tuesday, Duncan Dauvergne (Toronto) */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 7, '''Tuesday''' Van Vleck 901, 2:25pm,, Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17368Probability Seminar2019-04-22T18:25:44Z<p>Pmwood: /* Friday, April 26, Colloquium, B 239 from 4pm to 5pm, Kavita Ramanan, Brown */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, Van Vleck 911 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 7, '''Tuesday''', Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17346Probability Seminar2019-04-18T18:28:32Z<p>Pmwood: /* April 26, Colloquium, Kavita Ramanan, Brown */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== Friday, April 26, Colloquium, B 239 from 4pm to 5pm, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 7, '''Tuesday''', Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17345Probability Seminar2019-04-18T18:23:12Z<p>Pmwood: /* April 25, Kavita Ramanan, Brown */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs'''<br />
<br />
Abstract: Many applications can be modeled as a large system of homogeneous interacting particle systems on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle. This is based on various joint works with Ankan Ganguly, Dan Lacker and Ruoyu Wu.<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
Title: '''Tales of Random Projections'''<br />
<br />
Abstract: The interplay between geometry and probability in high-dimensional spaces is a subject of active research. Classical theorems in probability theory such as the central limit theorem and Cramer’s theorem can be viewed as providing information about certain scalar projections of high-dimensional product measures. In this talk we will describe the behavior of random projections of more general (possibly non-product) high-dimensional measures, which are of interest in diverse fields, ranging from asymptotic convex geometry to high-dimensional statistics. Although the study of (typical) projections of high-dimensional measures dates back to Borel, only recently has a theory begun to emerge, which in particular identifies the role of certain geometric assumptions that lead to better behaved projections. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been studied over the past decade, we describe more recent work on the tail behavior of multidimensional projections, and associated conditional limit theorems.<br />
<br />
== May 7, '''Tuesday''', Duncan Dauvergne (Toronto) ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17295Probability Seminar2019-04-09T15:39:09Z<p>Pmwood: /* April 18, Andrea Agazzi, Duke */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of processes.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
<!-- == April 26, TBA == --><br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17294Probability Seminar2019-04-09T15:35:06Z<p>Pmwood: /* April 18, Andrea Agazzi, Duke */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
'''Title: Stabilization of Diffusion Limited Aggregation in a Wedge.''' <br />
<br />
Abstract: We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA and questions about the number of arms, growth and dimension. I will present some conjectures and open problems.<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
<br />
Title: '''Large Deviations Theory for Chemical Reaction Networks'''<br />
<br />
Abstract:<br />
The microscopic dynamics of well-stirred networks of chemical reactions are modeled as jump Markov processes. At large volume, one may expect in this framework to have a straightforward application of large deviation theory. This is not at all true, for the jump rates of this class of models are typically neither globally Lipschitz, nor bounded away from zero, with both blowup and absorption as quite possible scenarios. In joint work with Amir Dembo and Jean-Pierre Eckmann, we utilize Lyapunov stability theory to bypass this challenges and to characterize a large class of network topologies that satisfy the full Wentzell-Freidlin theory of asymptotic rates of exit from domains of attraction. Under the assumption of positive recurrence these results also allow for the estimation of transitions times between metastable states of this class of proccesses.<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
<!-- == April 26, TBA == --><br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17243Probability Seminar2019-03-29T18:19:53Z<p>Pmwood: /* April 4, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, [https://www.math.wisc.edu/~pmwood/ Philip Matchett Wood], [http://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
<!-- == April 26, TBA == --><br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17242Probability Seminar2019-03-29T18:18:24Z<p>Pmwood: /* April 4, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
<!-- == April 26, TBA == --><br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17241Probability Seminar2019-03-29T18:18:14Z<p>Pmwood: /* April 4, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
<br />
Title: '''Outliers in the spectrum for products of independent random matrices'''<br />
<br />
Abstract: For fixed positive integers m, we consider the product of m independent n by n random matrices with iid entries as in the limit as n tends to infinity. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Terence Tao for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. Joint work with Natalie Coston and Sean O'Rourke.<br />
<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
<!-- == April 26, TBA == --><br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17069Probability Seminar2019-03-01T01:33:21Z<p>Pmwood: /* Probability related talk in PDE Geometric Analysis seminar: Monday, 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, February 22 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
<!-- == April 26, TBA == --><br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17040Probability Seminar2019-02-26T06:09:54Z<p>Pmwood: /* April 26, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
<!-- == April 26, TBA == --><br />
<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17039Probability Seminar2019-02-26T06:08:57Z<p>Pmwood: /* March 14, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
<!-- == March 14, TBA == --><br />
<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=17038Probability Seminar2019-02-26T06:08:43Z<p>Pmwood: /* March 7, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
<!-- == March 7, TBA == --><br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16985Probability Seminar2019-02-18T16:32:47Z<p>Pmwood: /* Probability related talk in PDE Geometric Analysis seminar: Monday, 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: <br> Monday, 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16984Probability Seminar2019-02-18T16:32:31Z<p>Pmwood: /* February 21, Diane Holcomb, KTH */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: Monday, 3:30pm to 4:30pm, Van Vleck 901, Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16983Probability Seminar2019-02-18T16:32:09Z<p>Pmwood: /* Probability related talk in PDE Geometric Analysis seminar: Monday, 3:30pm to 4:30pm, Van Vleck 901 */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: Monday, 3:30pm to 4:30pm, Van Vleck 901,<br />
Xiaoqin Guo, UW-Madison ==<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16982Probability Seminar2019-02-18T16:31:46Z<p>Pmwood: /* Wednesday, February 27 at 1:10pm Jon Peterson, Purdue */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== Probability related talk in PDE Geometric Analysis seminar: Monday, 3:30pm to 4:30pm, Van Vleck 901==<br />
<br />
Xiaoqin Guo, UW-Madison<br />
<br />
Title: Quantitative homogenization in a balanced random environment<br />
<br />
Abstract: Stochastic homogenization of discrete difference operators is closely related to the convergence of random walk in a random environment (RWRE) to its limiting process. In this talk we discuss non-divergence form difference operators in an i.i.d random environment and the corresponding process—a random walk in a balanced random environment in the integer lattice Z^d. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As consequences, we obtain algebraic rates of convergence for the quenched central limit theorem of the RWRE and for the homogenization of both elliptic and parabolic non-divergence form difference operators. Joint work with J. Peterson (Purdue) and H. V. Tran (UW-Madison).<br />
<br />
<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16981Probability Seminar2019-02-18T16:26:30Z<p>Pmwood: /* Wednesday, February 27 at 1:10pm Jon Peterson, Purdue */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
Title: '''Functional Limit Laws for Recurrent Excited Random Walks'''<br />
<br />
Abstract:<br />
<br />
Excited random walks (also called cookie random walks) are model for self-interacting random motion where the transition probabilities are dependent on the local time at the current location. While self-interacting random walks are typically very difficult to study, many results for (one-dimensional) excited random walks are remarkably explicit. In particular, one can easily (by hand) calculate a parameter of the model that will determine many features of the random walk: recurrence/transience, non-zero limiting speed, limiting distributions and more. In this talk I will prove functional limit laws for one-dimensional excited random walks that are recurrent. For certain values of the parameters in the model the random walks under diffusive scaling converge to a Brownian motion perturbed at its extremum. This was known previously for the case of excited random walks with boundedly many cookies per site, but we are able to generalize this to excited random walks with periodic cookie stacks. In this more general case, it is much less clear why perturbed Brownian motion should be the correct scaling limit. This is joint work with Elena Kosygina.<br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16911Probability Seminar2019-02-13T01:43:38Z<p>Pmwood: /* February 21, Diane Holcomb, KTH */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, [https://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], UW-Madison==<br />
<br />
Title: '''Geometry of the corner growth model'''<br />
<br />
Abstract: The corner growth model is a last-passage percolation model of random growth on the square lattice. It lies at the nexus of several branches of mathematics: probability, statistical physics, queueing theory, combinatorics, and integrable systems. It has been studied intensely for almost 40 years. This talk reviews properties of the geodesics, Busemann functions and competition interfaces of the corner growth model, and presents some new qualitative and quantitative results. Based on joint projects with Louis Fan (Indiana), Firas Rassoul-Agha and Chris Janjigian (Utah).<br />
<br />
== February 21, [https://people.kth.se/~holcomb/ Diane Holcomb], KTH ==<br />
<br />
<br />
Title: '''On the centered maximum of the Sine beta process'''<br />
<br />
<br />
Abstract: There has been a great deal or recent work on the asymptotics of the maximum of characteristic polynomials or random matrices. Other recent work studies the analogous result for log-correlated Gaussian fields. Here we will discuss a maximum result for the centered counting function of the Sine beta process. The Sine beta process arises as the local limit in the bulk of a beta-ensemble, and was originally described as the limit of a generalization of the Gaussian Unitary Ensemble by Valko and Virag with an equivalent process identified as a limit of the circular beta ensembles by Killip and Stoiciu. A brief introduction to the Sine process as well as some ideas from the proof of the maximum will be covered. This talk is on joint work with Elliot Paquette.<br />
<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the ''character ratio'': <br />
<br />
$$<br />
\text{trace}(\rho(g))/\text{dim}(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant ''rank''. This talk will discuss the notion of rank for $GL_n$ over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=16889Colloquia2019-02-09T21:04:16Z<p>Pmwood: /* Spring 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday, VV 911'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 6 '''Wednesday, room 911'''<br />
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)<br />
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs ]]<br />
| Anderson<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[# TBA| TBA ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Eviatar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Li-Cheng Tsai (Columbia University)===<br />
<br />
Title: When particle systems meet PDEs<br />
<br />
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16888Probability Seminar2019-02-09T21:02:59Z<p>Pmwood: /* April 11, Eviatar Proccia, Texas A&M */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
<!--== February 14, TBA ==--><br />
<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Procaccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16878Probability Seminar2019-02-08T17:55:25Z<p>Pmwood: /* February 14, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
<!--== February 14, TBA ==--><br />
<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16826Probability Seminar2019-02-05T01:53:29Z<p>Pmwood: /* March 28, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 901</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16825Probability Seminar2019-02-05T01:53:18Z<p>Pmwood: /* March 7, Shamgar Gurevitch UW-Madison */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 901</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16824Probability Seminar2019-02-05T01:51:40Z<p>Pmwood: /* March 7, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 901</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] ==<br />
<br />
Title: '''When particle systems meet PDEs'''<br />
<br />
Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems..<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, [https://www.math.wisc.edu/~shamgar/ Shamgar Gurevitch] [https://www.math.wisc.edu/ UW-Madison]==<br />
<br />
Title: '''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).<br />
<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16789Probability Seminar2019-01-30T16:57:16Z<p>Pmwood: /* 1:10pm Wednesday, February 27, Jon Peterson, Purdue */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27 at 1:10pm</span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16788Probability Seminar2019-01-30T16:56:07Z<p>Pmwood: /* Wednesday, February 27, Jon Peterson, Purdue */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> 1:10pm Wednesday, February 27, </span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16787Probability Seminar2019-01-30T16:55:48Z<p>Pmwood: /* Wednesday, February 27, Jon Peterson, Purdue */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27, </span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day and time. <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16786Probability Seminar2019-01-30T16:55:31Z<p>Pmwood: /* Wednesday, February 27, Jon Peterson, Purdue */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27, </span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:520px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day, time, <br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16785Probability Seminar2019-01-30T16:55:09Z<p>Pmwood: /* February 27, Jon Peterson, Purdue */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== <span style="color:red"> Wednesday, February 27, </span> [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
<br />
<div style="width:320px;height:50px;border:5px solid black"><br />
<b><span style="color:red">&emsp; Please note the unusual day, time, <br><br />
&emsp; </span></b><br />
</div><br />
<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16784Probability Seminar2019-01-30T16:53:52Z<p>Pmwood: /* February 28, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== February 27, [http://www.math.purdue.edu/~peterson/ Jon Peterson], [http://www.math.purdue.edu/ Purdue] ==<br />
<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_group_timetable&diff=16738Probability group timetable2019-01-27T01:57:58Z<p>Pmwood: </p>
<hr />
<div>2018 Fall<br />
<br />
<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Benedek 531 || || Benedek 531 || || Benedek 531 <br />
|- <br />
| 10-11|| Hao 431, Bedenk 833 || || Hao 431, Benedek 833 || || Hao 431, Benedek 833 <br />
|-<br />
| 11-12|| || || || ||<br />
|-<br />
| 12-1|| Phil 632 || || Phil 632 || || Phil 632 <br />
|-<br />
| 1-2|| || Timo 734 || || Timo 734 ||<br />
|-<br />
| 2-3|| Phil 632 || graduate probability seminar (2:25) || Phil 632 || probability seminar (2:25) || Phil 632<br />
|-<br />
| 3-4|| || || || || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
<br />
<br />
<!-- <br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Timo 431, Kurt 222|| Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431, Kurt 222 || Benedek 431, Sebastien 632, Louis 735 (9:30), Kurt CS719 || Timo 431<br />
|-<br />
| 10-11|| Kurt 222, Hans 234 || Phil out all day, Kurt 735 || Kurt 222, Hans 234 || Kurt 735 || Phil out all day, Hans 234 <br />
|-<br />
| 11-12|| Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Jinsu 375, Kurt 222, Hans 846, Christian 846 || Jinsu 375, Kurt 703 || Hans 846, Christian 846<br />
|-<br />
| 12-1|| Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431, Jinsu 375 || Kurt 703 (12:15) || Dave 431 <br />
|-<br />
| 1-2|| || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 || || Sebastien 632, Benedek 733, Jinsu 801, Hans 234 ||<br />
|-<br />
| 2-3|| Daniele 431 (2:25) || graduate probability seminar (2:25) || Daniele 431 (2:25) || probability seminar (2:25) || Daniele 431 (2:25)<br />
|-<br />
| 3-4|| || Kurt 222, Hans 234 || || Kurt 222, Hans 234 || <br />
|-<br />
| 4-5|| || || || || colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--><br />
<br />
<!--<br />
{| border="2"<br />
| ||Monday||Tuesday||Wednesday||Thursday||Friday<br />
|-<br />
| 9-10|| Phil out all day || Benedek 531 (9:30)|| || Benedek 531 (9:30) || Phil out all day<br />
|-<br />
| 10-11||Jinsu 722, Louis 431 || || Jinsu 722, Louis 431|| ||Jinsu 722, Louis 431<br />
|-<br />
| 11-12|| || Hans 820 || || Hans 820 ||<br />
|-<br />
| 12-1|| Jinsu 222, Louis 632 || ||Jinsu 222, Louis 632 || || Jinsu 222, Louis 632<br />
|-<br />
| 1-2|| Jinsu 222, Hans 851 || Benedek OH, Hans 843 || Jinsu 222, Hans 851|| Hans 843 ||Jinsu 222, Hans 851<br />
|-<br />
| 2-3|| || graduate probability seminar (2:25) || Louis (Seb) || probability seminar (2:25) ||<br />
|-<br />
| 3-4|| ||Benedek (OH (3:30) || Benedek OH || || <br />
|-<br />
| 4-5|| || || Louis (OH 4:30)|| Louis (OH 4:30)|| colloquium<br />
|-<br />
| 5-6|| || || || ||<br />
|}<br />
--></div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16731Probability Seminar2019-01-25T16:51:11Z<p>Pmwood: /* January 31, Oanh Nguyen, Princteton */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princeton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== February 28, TBA ==<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16717Probability Seminar2019-01-25T01:49:58Z<p>Pmwood: /* January 31, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, [https://www.math.princeton.edu/people/oanh-nguyen Oanh Nguyen], [https://www.math.princeton.edu/ Princteton] ==<br />
<br />
Title: '''Survival and extinction of epidemics on random graphs with general degrees'''<br />
<br />
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$.<br />
Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.<br />
<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== February 28, TBA ==<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
<br />
--><br />
<br />
== ==<br />
<br />
[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16618Probability Seminar2019-01-15T16:25:52Z<p>Pmwood: /* January 24, TBA */</p>
<hr />
<div>__NOTOC__<br />
<br />
= Spring 2019 =<br />
<br />
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
<br />
If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
<br />
<br />
<br />
== January 31, TBA ==<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
<br />
Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
<br />
Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
<br />
== February 14, TBA ==<br />
== February 21, TBA ==<br />
== February 28, TBA ==<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
<br />
== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
<br />
== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
<br />
== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
<br />
== April 26, TBA ==<br />
== May 2, TBA ==<br />
<br />
<br />
<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
<br />
<br />
Title: '''The distribution of sandpile groups of random regular graphs'''<br />
<br />
Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
<br />
Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
<br />
<br />
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
<br />
Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
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[[Past Seminars]]</div>Pmwoodhttps://www.math.wisc.edu/wiki/index.php?title=Probability_Seminar&diff=16609Probability Seminar2019-01-14T15:53:08Z<p>Pmwood: /* February 7, TBA */</p>
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<div>__NOTOC__<br />
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= Spring 2019 =<br />
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<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted. <br />
<b>We usually end for questions at 3:15 PM.</b><br />
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If you would like to sign up for the email list to receive seminar announcements then please send an email to <br />
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]<br />
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== January 24, TBA ==<br />
== January 31, TBA ==<br />
== February 7, [http://www.math.cmu.edu/~yug2/ Yu Gu], [https://www.cmu.edu/math/index.html CMU] ==<br />
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Title: '''Fluctuations of the KPZ equation in d\geq 2 in a weak disorder regime'''<br />
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Abstract: We will discuss some recent work on the Edwards-Wilkinson limit of the KPZ equation with a small coupling constant in d\geq 2.<br />
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== February 14, TBA ==<br />
== February 21, TBA ==<br />
== February 28, TBA ==<br />
== March 7, TBA ==<br />
== March 14, TBA ==<br />
== March 21, Spring Break, No seminar ==<br />
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== March 28, TBA ==<br />
== April 4, TBA ==<br />
== April 11, [https://sites.google.com/site/ebprocaccia/ Eviatar Proccia], [http://www.math.tamu.edu/index.html Texas A&M] ==<br />
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== April 18, [https://services.math.duke.edu/~agazzi/index.html Andrea Agazzi], [https://math.duke.edu/ Duke] ==<br />
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== April 25, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
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== April 26, Colloquium, [https://www.brown.edu/academics/applied-mathematics/kavita-ramanan Kavita Ramanan], [https://www.brown.edu/academics/applied-mathematics/ Brown] ==<br />
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== April 26, TBA ==<br />
== May 2, TBA ==<br />
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<!--<br />
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==<br />
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Title: '''The distribution of sandpile groups of random regular graphs'''<br />
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Abstract:<br />
We study the distribution of the sandpile group of random <math>d</math>-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the <math>p</math>-Sylow subgroup of the sandpile group is a given <math>p</math>-group <math>P</math>, is proportional to <math>|\operatorname{Aut}(P)|^{-1}</math>. For finitely many primes, these events get independent in limit. Similar results hold for undirected random regular graphs, there for odd primes the limiting distributions are the ones given by Clancy, Leake and Payne.<br />
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Our results extends a recent theorem of Huang saying that the adjacency matrices of random <math>d</math>-regular directed graphs are invertible with high probability to the undirected case.<br />
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==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==<br />
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Title: '''Stochastic quantization of Yang-Mills'''<br />
<br />
Abstract:<br />
"Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. Interesting progress has been made these years in understanding these SPDEs, examples including Phi4 and sine-Gordon. Yang-Mills is a type of quantum field theory which has gauge symmetry, and its stochastic quantization is a Yang-Mills flow perturbed by white noise.<br />
In this talk we start by an Abelian example where we take a symmetry-preserving lattice regularization and study the continuum limit. We will then discuss non-Abelian Yang-Mills theories and introduce a symmetry-breaking smooth regularization and restore the symmetry using a notion of gauge-equivariance. With these results we can construct dynamical Wilson loop and string observables. Based on [S., arXiv:1801.04596] and [Chandra,Hairer,S., work in progress].<br />
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--><br />
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== ==<br />
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[[Past Seminars]]</div>Pmwood