Difference between revisions of "Probability Seminar"

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(Thursday, May 7, Jessica Lin, UW-Madison, Room TBA)
(April 23, 2020, Martin Hairer (Imperial College))
 
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__NOTOC__
 
__NOTOC__
  
= Spring 2015 =
+
= Spring 2020 =
  
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
+
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.  
 +
<b>We  usually end for questions at 3:20 PM.</b>
  
<b>
+
If you would like to sign up for the email list to receive seminar announcements then please send an email to  
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu.
+
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]
  
<!-- [[File:probsem.jpg]] -->
 
</b>
 
 
= =
 
 
== Thursday, January 15, [http://www.stat.berkeley.edu/~racz/ Miklos Racz], [http://statistics.berkeley.edu/ UC-Berkeley Stats] ==
 
 
 
Title: Testing for high-dimensional geometry in random graphs
 
 
Abstract: I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels; we are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from an Erdos-Renyi random graph. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we use a bound on the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, and Ronen Eldan.
 
 
== Thursday, January 22, No Seminar  ==
 
 
== Thursday, January 29, [http://www.math.umn.edu/~arnab/ Arnab Sen], [http://www.math.umn.edu/ University of Minnesota]  ==
 
 
Title: '''Double Roots of Random Littlewood Polynomials'''
 
 
Abstract:
 
We consider random polynomials whose coefficients are independent and uniform on {-1,1}. We will show that the probability that such a polynomial of degree n has a double root is o(n^{-2}) when n+1 is not divisible by 4 and is of the order n^{-2} otherwise. We will also discuss extensions to random polynomials with more general coefficient distributions. 
 
 
This is joint work with Ron Peled and Ofer Zeitouni.
 
 
== Thursday, February 5, No seminar this week  ==
 
 
== Thursday, February 12, No Seminar this week==
 
 
 
<!--
 
== Wednesday, <span style="color:red">February 11</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison]  ==
 
 
<span style="color:red">Please note the unusual time and room.
 
</span>
 
 
 
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena
 
 
 
Abstract:
 
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.
 
--->
 
 
== Thursday, February 19, [http://www.math.purdue.edu/people/bio/guo297 Xiaoqin Guo], [http://www.math.purdue.edu/ Purdue]  ==
 
 
Title: Quenched invariance principle for random walks in time-dependent random environment
 
 
Abstract: In this talk we discuss random walks in a time-dependent zero-drift random environment in <math>Z^d</math>. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez.
 
 
== Thursday, February 26, [http://wwwf.imperial.ac.uk/~dcrisan/ Dan Crisan], [http://www.imperial.ac.uk/natural-sciences/departments/mathematics/ Imperial College London]  ==
 
 
Title: '''Smoothness properties of randomly perturbed semigroups with application to nonlinear filtering'''
 
 
Abstract:
 
In this talk I will discuss sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. The estimates we derive have sharp small time asymptotics
 
 
   
 
   
This is joint work with Terry Lyons (Oxford) and Christian Literrer (Ecole Polytechnique) and is based on the paper
+
== January 23, 2020, [https://www.math.wisc.edu/~seppalai/ Timo Seppalainen] (UW Madison) ==
 
+
'''Non-existence of bi-infinite geodesics in the exponential corner growth model
D Crisan, C Litterer, T Lyons, Kusuoka–Stroock gradient bounds for the solution of the filtering equation, Journal of Functional Analysis, 2105
+
'''
 
 
== Wednesday, <span style="color:red">March 4</span>, [http://www.math.wisc.edu/~stechmann/ Sam Stechmann], [http://www.math.wisc.edu/ UW-Madison], <span style="color:red"> 2:25pm Van Vleck B113</span>  ==
 
 
 
<span style="color:red">Please note the unusual time and room.
 
</span>
 
 
 
 
 
Title: Stochastic Models for Rainfall: Extreme Events and Critical Phenomena
 
 
 
 
 
Abstract:
 
In recent years, tropical rainfall statistics have been shown to conform to paradigms of critical phenomena and statistical physics. In this talk, stochastic models will be presented as prototypes for understanding the atmospheric dynamics that leads to these statistics and extreme events. Key nonlinear ingredients in the models include either stochastic jump processes or thresholds (Heaviside functions). First, both exact solutions and simple numerics are used to verify that a suite of observed rainfall statistics is reproduced by the models, including power-law distributions and long-range correlations. Second, we prove that a stochastic trigger, which is a time-evolving indicator of whether it is raining or not, will converge to a deterministic threshold in an appropriate limit. Finally, we discuss the connections among these rainfall models, stochastic PDEs, and traditional models for critical phenomena.
 
 
 
== Thursday, March 12, [http://www.ima.umn.edu/~ohadfeld/Website/index.html Ohad Feldheim], [http://www.ima.umn.edu/ IMA]  ==
 
 
 
 
 
Title: '''The 3-states AF-Potts model in high dimension'''
 
 
 
Abstract:
 
<!--
 
Take a bounded odd domain of the bipartite graph $\mathbb{Z}^d$. Color the boundary of the set by $0$, then
 
color the rest of the domain at random with the colors $\{0,\dots,q-1\}$, penalizing every
 
configuration with proportion to the number of improper edges at a given rate $\beta>0$ (the "inverse temperature").
 
Q: "What is the structure of such a coloring?"
 
 
 
This model is called the $q$-states Potts antiferromagnet(AF), a classical spin glass model in statistical mechanics.
 
The $2$-states case is the famous Ising model which is relatively well understood.
 
The $3$-states case in high dimension has been studies for $\beta=\infty$,
 
when the model reduces to a uniformly chosen proper three coloring of the domain.
 
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the structure of the model
 
showing long-range correlations and phase coexistence. In this work, we generalize this result to positive temperature,
 
showing that for large enough $\beta$ (low enough temperature)
 
the rigid structure persists. This is the first rigorous result for $\beta<\infty$.
 
 
 
In the talk, assuming no acquaintance with the model, we shall give the physical background, introduce all the
 
relevant definitions and shed some light on how such results are proved using only combinatorial methods.
 
Joint work with Yinon Spinka.
 
-->
 
Take a bounded odd domain of the bipartite graph <math>\mathbb{Z}^d</math>.    Color the
 
boundary of the set by <math>0</math>, then
 
color the rest of the domain at random with the colors <math>\{0,\dots,q-1\}</math>,
 
penalizing every
 
configuration with proportion to the number of improper edges at a given rate
 
<math>\beta>0</math> (the "inverse temperature").
 
Q: "What is the structure of such a coloring?"
 
 
 
This model is called the <math>q</math>-states Potts antiferromagnet(AF), a    classical spin
 
glass model in statistical mechanics.
 
The <math>2</math>-states case is the famous Ising model which is relatively    well
 
understood.
 
The <math>3</math>-states case in high dimension has been studies for          <math>\beta=\infty</math>,
 
when the model reduces to a uniformly chosen proper three coloring of the
 
domain.
 
Several words, by Galvin, Kahn, Peled, Randall and Sorkin established the
 
structure of the model
 
showing long-range correlations and phase coexistence. In this work, we
 
generalize this result to positive temperature,
 
showing that for large enough <math>\beta</math> (low enough temperature)
 
the rigid structure persists. This is the first rigorous result for
 
<math>\beta<\infty</math>.
 
 
 
In the talk, assuming no acquaintance with the model, we shall give the
 
physical background, introduce all the
 
relevant definitions and shed some light on how such results are proved using
 
only combinatorial methods.
 
Joint work with Yinon Spinka.
 
 
 
== Thursday, March 19,  [http://www.cmc.edu/pages/faculty/MHuber/ Mark Huber], [http://www.cmc.edu/math/ Claremont McKenna Math]  ==
 
 
 
Title: Understanding relative error in Monte Carlo simulations
 
 
 
Abstract:  The problem of estimating the probability <math>p</math> of heads on an unfair coin has been around for centuries, and has inspired numerous advances in probability such as the Strong Law of Large Numbers and the Central Limit Theorem.  In this talk, I'll consider a new twist:  given an estimate <math>\hat p</math>, suppose we want to understand the behavior of the relative error <math>(\hat p - p)/p</math>.  In classic estimators, the values that the relative error can take on depends on the value of <math>p</math>.  I will present a new estimate with the remarkable property that the distribution of the relative error does not depend in any way on the value of <math>p</math>.  Moreover, this new estimate is very fast:  it takes a number of coin flips that is very close to the theoretical minimum.  Time permitting, I will also discuss new ways to use concentration results for estimating the mean of random variables where normal approximations do not apply.
 
 
 
== Thursday, March 26, [http://mathsci.kaist.ac.kr/~jioon/ Ji Oon Lee], [http://www.kaist.edu/html/en/index.html KAIST]  ==
 
 
 
Title: Tracy-Widom Distribution for Sample Covariance Matrices with General Population
 
 
 
Abstract:
 
Consider the sample covariance matrix <math>(\Sigma^{1/2} X)(\Sigma^{1/2} X)^*</math>, where the sample <math>X</math> is an <math>M \times N</math>  random matrix whose entries are real independent random variables with variance <math>1/N</math> and <math>\Sigma</math> is an  <math>M \times M</math> positive-definite deterministic diagonal matrix. We show that the fluctuation of its rescaled largest eigenvalue is given by the type-1 Tracy-Widom distribution. This is a joint work with Kevin Schnelli.
 
 
 
== Thursday, April 2, No Seminar, Spring Break  ==
 
 
 
 
 
 
 
 
 
== Thursday, April 9, [http://www.math.wisc.edu/~emrah/ Elnur Emrah], [http://www.math.wisc.edu/ UW-Madison]  ==
 
 
 
Title: The shape functions of certain exactly solvable inhomogeneous planar corner growth models
 
 
 
Abstract: I will talk about two kinds of inhomogeneous corner growth models with independent waiting times {W(i, j): i, j positive integers}: (1) W(i, j) is distributed exponentially with parameter <math>a_i+b_j</math> for each i, j.(2) W(i, j) is distributed geometrically with fail parameter <math>a_ib_j</math> for each i, j. These generalize exactly-solvable i.i.d. models with exponential or geometric waiting times. The parameters (a_n) and (b_n) are random with a joint distribution that is stationary with respect to the nonnegative shifts and ergodic (separately) with respect to the positive shifts of the indices. Then the shape functions of models (1) and (2) satisfy variational formulas in terms of the marginal distributions of (a_n) and (b_n). For certain choices of these marginal distributions, we still get closed-form expressions for the shape function as in the i.i.d. models.
 
 
 
== Thursday, April 16, [http://www.math.wisc.edu/~shottovy/ Scott Hottovy], [http://www.math.wisc.edu/ UW-Madison] ==
 
 
 
Title: '''An SDE approximation for stochastic differential delay equations with colored state-dependent noise'''
 
 
 
Abstract: In this talk I will introduce a stochastic differential delay equation with state-dependent colored noise which arises from a noisy circuit experiment. In the experimental paper, a small delay and correlation time limit was performed by using a Taylor expansion of the delay. However, a time substitution was first performed to obtain a good match with experimental results. I will discuss how this limit can be proved without the use of a Taylor expansion by using a theory of convergence of stochastic processes developed by Kurtz and Protter. To obtain a necessary bound, the theory of sums of weakly dependent random variables is used. This analysis leads to the explanation of why the time substitution was needed in the previous case.
 
 
 
== Thursday, April 23, [http://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [http://math.osu.edu/ Ohio State University]  ==
 
 
 
Title: On eigenvalue repulsion of random matrices
 
 
 
Abstract:
 
 
 
I will address certain repulsion behavior of roots of random polynomials and of eigenvalues of Wigner matrices, and their applications. Among other things, we show a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for quite general matrix ensembles.
 
 
 
 
 
== Thursday, May 7, [http://www.math.wisc.edu/~jessica/ Jessica Lin], [http://www.math.wisc.edu/ UW-Madison], <span style="color:red"> 2:25pm, Room TBA </span>  ==
 
 
 
<span style="color:red">Please note the unusual room, which is to be announced.
 
</span>
 
 
 
 
 
 
 
Title: '''Random Walks in Random Environments and Stochastic Homogenization'''
 
 
 
In this talk, I will draw connections between random walks in random environments (RWRE) and stochastic homogenization of partial differential equations (PDE). I will introduce various models of RWRE and derive the corresponding PDEs to show that the two subjects are intimately related. I will then give a brief overview of the tools and techniques used in both approaches (reviewing some classical results), and discuss some recent problems in RWRE which are related to my research in stochastic homogenization.
 
 
 
== Thursday, May 14, [http://www.math.wisc.edu/~janjigia/ Chris Janjigian], [http://www.math.wisc.edu UW-Madison]  ==
 
 
 
Title: '''Large deviations of the free energy in the O’Connell-Yor polymer'''
 
 
 
Abstract: The first model of a directed polymer in a random environment was introduced in the statistical physics literature in the mid 1980s. This family of models has attracted substantial interest in the mathematical community in recent years, due in part to the conjecture that they lie in the KPZ universality class. At the moment, this conjecture can only be verified rigorously for a handful of exactly solvable models. In order to further explore the behavior of these models, it is natural to question whether the solvable models have any common features aside from the Tracy-Widom fluctuations and scaling exponents that characterize the KPZ class.
 
 
 
This talk considers the behavior of one of the solvable polymer models when it is far away from the behavior one would expect based on the KPZ conjecture. We consider the model of a 1+1 dimensional directed polymer model due to O’Connell and Yor, which is a Brownian analogue of the classical lattice polymer models. This model satisfies a strong analogue of Burke’s theorem from queueing theory, which makes some objects of interest computable. This talk will discuss how, using the Burke property, one can compute the positive moment Lyapunov exponents of the parabolic Anderson model associated to the polymer and how this leads to a computation of the large deviation rate function with normalization n for the free energy of the polymer.
 
 
 
 
 
 
 
<!--
 
== Thursday, December 11, TBA  ==
 
 
 
Title: TBA
 
 
 
Abstract:
 
-->
 
 
 
 
 
 
 
<!--
 
 
 
== Thursday, September 11, <span style="color:red">Van Vleck B105,</span> [http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood], UW-Madison ==
 
 
 
Please note the non-standard room.
 
 
 
Title: '''The distribution of sandpile groups of random graphs'''
 
 
 
Abstract:<br>
 
The sandpile group is an abelian group associated to a graph, given as
 
the cokernel of the graph Laplacian.  An Erdős–Rényi random graph
 
then gives some distribution of random abelian groups.  We will give
 
an introduction to various models of random finite abelian groups
 
arising in number theory and the connections to the distribution
 
conjectured by Payne et. al. for sandpile groups.  We will talk about
 
the moments of random finite abelian groups, and how in practice these
 
are often more accessible than the distributions themselves, but
 
frustratingly are not a priori guaranteed to determine the
 
distribution.  In this case however, we have found the moments of the
 
sandpile groups of random graphs, and proved they determine the
 
measure, and have proven Payne's conjecture.
 
 
 
== Thursday, September 18, [http://www.math.purdue.edu/~peterson/ Jonathon Peterson], [http://www.math.purdue.edu/ Purdue University]  ==
 
 
 
Title: '''Hydrodynamic limits for directed traps and systems of independent RWRE'''
 
 
 
Abstract:
 
 
 
We study the evolution of a system of independent random walks in a common random environment (RWRE). Previously a hydrodynamic limit was proved in the case where the environment is such that the random walks are ballistic (i.e., transient with non-zero speed <math>v_0 \neq 0)</math>. In this case it was shown that the asymptotic particle density is simply translated deterministically by the speed $v_0$. In this talk we will consider the more difficult case of RWRE that are transient but with $v_0=0$. Under the appropriate space-time scaling, we prove a hydrodynamic limit for the system of random walks. The statement of the hydrodynamic limit that we prove is non-standard in that the evolution of the asymptotic particle density is given by the solution of a random rather than a deterministic PDE. The randomness in the PDE comes from the fact that under the hydrodynamic scaling the effect of the environment does not ``average out'' and so the specific instance of the environment chosen actually matters.
 
 
 
The proof of the hydrodynamic limit for the system of RWRE will be accomplished by coupling the system of RWRE with a simpler model of a system of particles in an environment of ``directed traps.'' This talk is based on joint work with Milton Jara.
 
 
 
== Thursday, September 25, [http://math.colorado.edu/~seor3821/ Sean O'Rourke],  [http://www.colorado.edu/math/ University of Colorado Boulder]  ==
 
 
 
Title: '''Singular values and vectors under random perturbation'''
 
 
 
Abstract:
 
Computing the singular values and singular vectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following. How much does a small perturbation to the matrix change the singular values and vectors?
 
  
Classical (deterministic) theorems, such as those by Davis-Kahan, Wedin, and Weyl, give tight estimates for the worst-case scenario. In this talk, I will consider the case when the perturbation is random. In this setting, better estimates can be achieved when our matrix has low rank. This talk is based on joint work with Van Vu and Ke Wang.
+
Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s.   A non-existence proof  in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in  November 2018. Their proof utilizes estimates from integrable probability.   This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).
  
== Thursday, October 2, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison==
+
== January 30, 2020, [https://www.math.wisc.edu/people/vv-prof-directory Scott Smith] (UW Madison) ==
 +
'''Quasi-linear parabolic equations with singular forcing'''
  
Title: '''Anisotropic local laws for random matrices'''
+
The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral.  By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise.  In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise.  The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.
  
Abstract:
+
In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solutionThese are known as quasi-linear equations.  Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization.  Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.
In this talk, we introduce a new method of deriving  local laws of random matricesAs applications, we will show the local laws  and some universality results on general sample covariance matrices: TXX^*T^* (where $T$ is non-square deterministic matrix), and deformed Wigner matrix: H+A (where A is deterministic symmetric matrix). Note: here $TT^*$ and $A$ could be full rank matrices.
 
  
== Thursday, October 9, No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium] ==
+
== February 6, 2020, [https://sites.google.com/site/cyleeken/ Cheuk-Yin Lee] (Michigan State) ==
 +
'''Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points'''
  
No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium].
+
In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.
  
 +
== February 13, 2020, [http://www.jelena-diakonikolas.com/ Jelena Diakonikolas] (UW Madison) ==
 +
'''Langevin Monte Carlo Without Smoothness'''
  
== Thursday, October 16, [http://www.math.utah.edu/~firas/ Firas Rassoul-Agha], [http://www.math.utah.edu/ University of Utah]==
+
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation.
 +
Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.
  
Title: '''The growth model: Busemann functions, shape, geodesics, and other stories'''
+
== February 20, 2020, [https://math.berkeley.edu/~pmwood/ Philip Matchett Wood] (UC Berkeley) ==
 +
'''A replacement principle for perturbations of non-normal matrices'''
  
Abstract:
+
There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added.  However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added. Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measureInterestingly, this even allows for deterministic perturbations to be considered.  Joint work with Sean O'Rourke.
We consider the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interfaceThis is joint work with Nicos Georgiou and Timo Seppalainen.
 
  
 +
== February 27, 2020, No seminar ==
 +
''' '''
  
== Thursday, November 6, Vadim Gorin, [http://www-math.mit.edu/people/profile.php?pid=1415 MIT] ==
+
== March 5, 2020, [https://www.ias.edu/scholars/jiaoyang-huang Jiaoyang Huang] (IAS) ==
 +
''' Large Deviation Principles via Spherical Integrals'''
  
Title: '''Multilevel Dyson Brownian Motion and its edge limits.'''
+
In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain
  
Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of
+
1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;
interacting particle systems on one side, and the asymptotic behavior for largest eigenvalues of
 
random Hermitian matrices on the other side. In my talk I will explain some reasons for this
 
connection between two seemingly unrelated classes of stochastic systems, and how this relation can
 
be extended to general beta random matrices. A multilevel extension of the Dyson Brownian Motion
 
will be the central object in the discussion.
 
  
(Based on joint papers with Misha Shkolnikov.)
+
2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;
  
==<span style="color:red"> Friday</span>, November 7, [http://tchumley.public.iastate.edu/ Tim Chumley], [http://www.math.iastate.edu/ Iowa State University] ==
+
3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;
  
<span style="color:darkgreen">Please note the unusual day.</span>
+
4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.
  
Title: '''Random billiards and diffusion'''
+
This is a joint work with Belinschi and Guionnet.
  
Abstract: We introduce a class of random dynamical systems derived from billiard maps and study a certain Markov chain derived from them. We then discuss the interplay between the billiard geometry and stochastic properties of the random system.  The main results presented investigate the affect of billiard geometry on a diffusion process obtained from an appropriate scaling limit of the Markov chain.
+
== March 12, 2020, No seminar ==
 +
''' '''
  
== Thursday, November 13, [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], [http://www.math.wisc.edu/ UW-Madison]==
+
== March 19, 2020, Spring break ==
 +
''' '''
  
Title: '''Variational formulas for directed polymer and percolation models'''
+
== March 26, 2020, CANCELLED, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
 +
''' '''
  
Abstract:
+
== April 2, 2020, CANCELLED, [http://pages.cs.wisc.edu/~tl/ Tianyu Liu] (UW Madison)==
Explicit formulas for subadditive limits of polymer and percolation models in probability and statistical mechanics have been difficult to find. We describe variational formulas for these limits and their connections with other features of the models such as Busemann functions and Kardar-Parisi-Zhang (KPZ) fluctuation exponents.
+
''' '''
  
 +
== April 9, 2020, CANCELLED, [http://stanford.edu/~ajdunl2/ Alexander Dunlap] (Stanford) ==
 +
''' '''
  
 +
== April 16, 2020, CANCELLED, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
 +
''' '''
  
== <span style="color:red">Monday</span>, December 1, [http://www.ma.utexas.edu/users/jneeman/index.html Joe Neeman], [http://www.ma.utexas.edu/ UT-Austin], <span style="color:red">4pm, Room B239 Van Vleck Hall</span>==
+
== April 22-24, 2020, CANCELLED, [http://frg.int-prob.org/ FRG Integrable Probability] meeting ==
  
<span style="color:darkgreen">Please note the unusual time and room.</span>
+
3-day event in Van Vleck 911
  
Title: '''Some phase transitions in the stochastic block model'''
+
== April 23, 2020, CANCELLED, [http://www.hairer.org/ Martin Hairer] (Imperial College) ==
  
Abstract: The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.
+
[https://www.math.wisc.edu/wiki/index.php/Colloquia Wolfgang Wasow Lecture] at 4pm in Van Vleck 911
  
== Thursday, December 4, Arjun Krishnan, [http://www.fields.utoronto.ca/ Fields Institute] ==
+
== April 30, 2020, [http://willperkins.org/ Will Perkins] (University of Illinois at Chicago) ==
 +
''' '''
  
Title: '''Variational formula for the time-constant of first-passage percolation'''
 
  
Abstract:
 
Consider first-passage percolation with positive, stationary-ergodic
 
weights on the square lattice in d-dimensions. Let <math>T(x)</math> be the
 
first-passage time from the origin to <math>x</math> in <math>Z^d</math>. The convergence of
 
<math>T([nx])/n</math> to the time constant as <math>n</math> tends to infinity is a consequence
 
of the subadditive ergodic theorem. This convergence can be viewed as
 
a problem of homogenization for a discrete Hamilton-Jacobi-Bellman
 
(HJB) equation. By borrowing several tools from the continuum theory
 
of stochastic homogenization for HJB equations, we derive an exact
 
variational formula (duality principle) for the time-constant. Under a
 
symmetry assumption, we will use the variational formula to construct
 
an explicit iteration that produces the limit shape.
 
  
  
-->
 
  
== ==
 
  
  
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 13:51, 24 March 2020


Spring 2020

Thursdays in 901 Van Vleck Hall at 2:30 PM, unless otherwise noted. We usually end for questions at 3:20 PM.

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu


January 23, 2020, Timo Seppalainen (UW Madison)

Non-existence of bi-infinite geodesics in the exponential corner growth model

Whether bi-infinite geodesics exist has been a significant open problem in first- and last-passage percolation since the mid-80s. A non-existence proof in the case of directed planar last-passage percolation with exponential weights was posted by Basu, Hoffman and Sly in November 2018. Their proof utilizes estimates from integrable probability. This talk describes an independent proof completed 10 months later that relies on couplings, coarse graining, and control of geodesics through planarity and increment-stationary last-passage percolation. Joint work with Marton Balazs and Ofer Busani (Bristol).

January 30, 2020, Scott Smith (UW Madison)

Quasi-linear parabolic equations with singular forcing

The classical solution theory for stochastic ODE's is centered around Ito's stochastic integral. By intertwining ideas from analysis and probability, this approach extends to many PDE's, a canonical example being multiplicative stochastic heat equations driven by space-time white noise. In both the ODE and PDE settings, the solution theory is beyond the scope of classical deterministic theory because of the ambiguity in multiplying a function with a white noise. The theory of rough paths and regularity structures provides a more quantitative understanding of this difficulty, leading to a more refined solution theory which efficiently divides the analytic and probabilistic aspects of the problem, and remarkably, even has an algebraic component.

In this talk, we will discuss a new application of these ideas to stochastic heat equations where the strength of the diffusion is not constant but random, as it depends locally on the solution. These are known as quasi-linear equations. Our main result yields the deterministic side of a solution theory for these PDE's, modulo a suitable renormalization. Along the way, we identify a formally infinite series expansion of the solution which guides our analysis, reveals a nice algebraic structure, and encodes the counter-terms in the PDE. This is joint work with Felix Otto, Jonas Sauer, and Hendrik Weber.

February 6, 2020, Cheuk-Yin Lee (Michigan State)

Sample path properties of stochastic partial differential equations: modulus of continuity and multiple points

In this talk, we will discuss sample path properties of stochastic partial differential equations (SPDEs). We will present a sharp regularity result for the stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. We prove the exact modulus of continuity via the property of local nondeterminism. We will also discuss the existence problem for multiple points (or self-intersections) of the sample paths of SPDEs. Our result shows that multiple points do not exist in the critical dimension for a large class of Gaussian random fields including the solution of a linear system of stochastic heat or wave equations.

February 13, 2020, Jelena Diakonikolas (UW Madison)

Langevin Monte Carlo Without Smoothness

Langevin Monte Carlo (LMC) is an iterative algorithm used to generate samples from a distribution that is known only up to a normalizing constant. The nonasymptotic dependence of its mixing time on the dimension and target accuracy is understood mainly in the setting of smooth (gradient-Lipschitz) log-densities, a serious limitation for applications in machine learning. We remove this limitation by providing polynomial-time convergence guarantees for a variant of LMC in the setting of non-smooth log-concave distributions. At a high level, our results follow by leveraging the implicit smoothing of the log-density that comes from a small Gaussian perturbation that we add to the iterates of the algorithm and while controlling the bias and variance that are induced by this perturbation. Based on joint work with Niladri Chatterji, Michael I. Jordan, and Peter L. Bartlett.

February 20, 2020, Philip Matchett Wood (UC Berkeley)

A replacement principle for perturbations of non-normal matrices

There are certain non-normal matrices whose eigenvalues can change dramatically when a small perturbation is added. However, when that perturbation is an iid random matrix, it appears that the eigenvalues become stable after perturbation and only change slightly when further small perturbations are added. Much of the work is this situation has focused on iid random gaussian perturbations. In this talk, we will discuss work on a universality result that allows for consideration of non-gaussian perturbations, and that shows that all perturbations satisfying certain conditions will produce the same limiting eigenvalue measure. Interestingly, this even allows for deterministic perturbations to be considered. Joint work with Sean O'Rourke.

February 27, 2020, No seminar

March 5, 2020, Jiaoyang Huang (IAS)

Large Deviation Principles via Spherical Integrals

In this talk, I'll explain a framework to study the large deviation principle for matrix models and their quantized versions, by tilting the measures using the asymptotics of spherical integrals obtained by Guionnet and Zeitouni. As examples, we obtain

1) the large deviation principle for the empirical distribution of the diagonal entries of $UB_NU^*$, for a sequence of $N\times N$ diagonal matrices $B_N$ and unitary/orthogonal Haar distributed matrices $U$;

2) the large deviation upper bound for the empirical eigenvalue distribution of $A_N+UB_NU^*$, for two sequences of $N\times N$ diagonal matrices $A_N, B_N$, and their complementary lower bounds at "good" probability distributions;

3) the large deviation principle for the Kostka number $K_{\lambda_N \eta_N}$, for two sequences of partitions $\lambda_N, \eta_N$ with at most $N$ rows;

4) the large deviation upper bound for the Littlewood-Richardson coefficients $c_{\lambda_N \eta_N}^{\kappa_N}$, for three sequences of partitions $\lambda_N, \eta_N, \kappa_N$ with at most $N$ rows, and their complementary lower bounds at "good" probability distributions.

This is a joint work with Belinschi and Guionnet.

March 12, 2020, No seminar

March 19, 2020, Spring break

March 26, 2020, CANCELLED, Philippe Sosoe (Cornell)

April 2, 2020, CANCELLED, Tianyu Liu (UW Madison)

April 9, 2020, CANCELLED, Alexander Dunlap (Stanford)

April 16, 2020, CANCELLED, Jian Ding (University of Pennsylvania)

April 22-24, 2020, CANCELLED, FRG Integrable Probability meeting

3-day event in Van Vleck 911

April 23, 2020, CANCELLED, Martin Hairer (Imperial College)

Wolfgang Wasow Lecture at 4pm in Van Vleck 911

April 30, 2020, Will Perkins (University of Illinois at Chicago)





Past Seminars