Past Probability Seminars Spring 2020: Difference between revisions

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= Spring 2015 =
= Fall 2018 =


<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
<b>Thursdays in 901 Van Vleck Hall at 2:25 PM</b>, unless otherwise noted.  
<b>We  usually end for questions at 3:15 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] ==
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==




Title: Testing for high-dimensional geometry in random graphs
Title: '''The distribution of sandpile groups of random regular 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:
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.
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.


This is joint work with Ron Peled and Ofer Zeitouni.
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.


== Thursday, February 5, No seminar this week  ==
<!-- ==September 13, TBA == -->


== Thursday, February 12, No Seminar this week==
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==
 
 
<!--
== 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


Title: '''Stochastic quantization of Yang-Mills'''


Abstract:
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.
"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.
--->
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].
 
== 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'''
==September 27, [https://www.math.wisc.edu/~seppalai/  Timo Seppäläinen] [https://www.math.wisc.edu/ UW-Madison] ==


Abstract:
Title:'''Random walk in random environment and the Kardar-Parisi-Zhang class'''
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
Abstract:This talk concerns a relationship between two much-studied classes of models  of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3.   Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah).
 
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] ==
==October 4, [https://people.math.osu.edu/paquette.30/ Elliot Paquette], [https://math.osu.edu/ OSU] ==


Title: TBA
Title: '''Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble'''


Abstract:
Abstract:
The characteristic polynomial of the Gaussian beta--ensemble can be represented, via its tridiagonal model, as an entry in a product of independent random two--by--two matrices.  For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane.  Conjecturally, we show that the characteristic polynomial is always represented as product of at most three terms, an exponential of a Gaussian field, the stochastic Airy function, and a diffusion similar to the stochastic sine equation.
We explain the origins of this decomposition, and we show partial progress in establishing part of it.


Joint work with Diane Holcomb and Gaultier Lambert.


== Thursday, April 2, No Seminar, Spring Break  ==
==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==
 




Title: '''Busemann functions and Gibbs measures in directed polymer models on Z^2'''


== Thursday, April 9, [http://www.math.wisc.edu/~emrah/ Elnur Emrah], [http://www.math.wisc.edu/ UW-Madison]  ==
Abstract: We consider the model of a nearest-neighbor random walk on the planar square lattice in a general iid space-time potential, which is also known as a directed polymer in a random environment. We prove results on existence, uniqueness (and non-uniqueness), and the law of large numbers for semi-infinite path measures. Our main tools are the Busemann functions, which are families of stochastic processes obtained through limits of ratios of partition functions.


Title: TBA
Based on joint work with Firas Rassoul-Agha


Abstract:
==October 18-20, [http://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium], No Seminar ==


==October 25, [http://stat.columbia.edu/department-directory/name/promit-ghosal/ Promit Ghosal], Columbia ==


== Thursday, April 16, TBA  ==


Title: TBA
Title: '''Tails of the KPZ equation'''
     
Abstract: The KPZ equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. It is related to another important SPDE, namely, the stochastic heat equation (SHE). In this talk, we focus on the tail probabilities of the solution of the KPZ equation. For instance, we investigate the probability of the solution being smaller or larger than the expected value. Our analysis is based on an exact identity between the KPZ equation and the Airy point process (which arises at the edge of the spectrum of the random Hermitian matrices) and the Brownian Gibbs property of the KPZ line ensemble.


Abstract:
This talk will be based on a joint work with my advisor Prof. Ivan Corwin.


== Thursday, April 23, [http://people.math.osu.edu/nguyen.1261/ Hoi Nguyen], [http://math.osu.edu/ Ohio State University] ==
==November 1, [https://math.umn.edu/directory/james-melbourne James Melbourne], [https://math.umn.edu/ University of Minnesota] ==


Title: TBA
Title: '''Upper bounds on the density of independent vectors under certain linear mappings'''


Abstract:
Abstract: Using functional analytic techniques and rearrangement, we prove anti-concentration results for the linear images of independent random variables, in the form of density upper bounds.  For continuous variables the results unify and sharpen Bobkov-Chistyakov's for independent sums of vectors and Rudelson-Vershynin's bounds on projections of independent coordinates.  For integer valued variables the techniques reduce finding the maximum of the probability mass function of a sum of independent variables, to the case that each variable is uniform on a contiguous interval.  This problem is approached through analysis of characteristic functions and new $L^p$ bounds on the Dirichlet and Fejer Kernel are obtained and used to derive a discrete analog of Bobkov-Chistyakov.


== Thursday, April 30, TBA  ==
==November 8, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], [https://cims.nyu.edu/ NYU] ==


Title: TBA
Title: '''The Sine-beta process: DLR equations and applications'''


Abstract:
Abstract:
One-dimensional log-gases, or Beta-ensembles, are statistical physics models finding an incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE's. In a joint work with D. Dereudre, A. Hardy, and M. Maïda, we give a new description of Sine-beta as an "infinite volume Gibbs measure", using the Dobrushin-Lanford-Ruelle (DLR) formalism, and we use it to prove the rigidity of the process, in the sense of Ghosh-Peres. Another application is a CLT for fluctuations of linear statistics.


<!-- ==November 15, TBA == -->


== Thursday, May 7, TBA  ==
==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==


Title: TBA
==  <span style="color:red">  Monday, November 26, 4pm, Van Vleck 911</span>  [http://math.mit.edu/directory/profile.php?pid=1415 Vadim Gorin], [http://math.mit.edu/index.php MIT]  ==


Abstract:


<div style="width:320px;height:50px;border:5px solid black">
<b><span style="color:red">&emsp; Please note the unusual day, time, <br>
&emsp; and room.</span></b>
</div>


 
Title: '''Macroscopic fluctuations through Schur generating functions'''
 
 
 
<!--
== Thursday, December 11, TBA  ==
 
Title: TBA


Abstract:
Abstract:
-->
I will talk about a special class of large-dimensional stochastic systems with
strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices,
and measures governing decompositions of group representations into irreducible components.


It is believed that macroscopic fluctuations in such systems are universally
described by log-correlated Gaussian fields. I will present an approach to
handle this question based on the notion of the Schur generating function of a probability
distribution, and explain how it leads to a rigorous confirmation of this belief in
a variety of situations.


<!-- ==November 29, TBA == -->


<!--
== <span style="color:red">  Wednesday, December 5 at 4pm in Van Vleck 911</span> [http://www.mit.edu/~ssen90/ Subhabrata Sen], [https://math.mit.edu/ MIT] and [https://www.microsoft.com/en-us/research/lab/microsoft-research-new-england/ Microsoft Research New England] ==


== 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.
<div style="width:320px;height:50px;border:5px solid black">
<b><span style="color:red">&emsp; Please note the unusual day, time, <br>&emsp; and room. </span></b>
</div>


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: '''Random graphs, Optimization, and Spin glasses'''
 
Title: '''Hydrodynamic limits for directed traps and systems of independent RWRE'''


Abstract:
Abstract:
Combinatorial optimization problems are ubiquitous in diverse mathematical
applications. The desire to understand their “typical” behavior motivates
a study of these problems on random instances. In spite of a long and rich
history, many natural questions in this domain are still intractable to rigorous
mathematical analysis. Graph cut problems such as Max-Cut and Min-bisection
are canonical examples in this class. On the other hand, physicists study these
questions using the non-rigorous “replica” and “cavity” methods, and predict
complex, intriguing features. In this talk, I will describe some recent progress
in our understanding of their typical properties on random graphs, obtained via
connections to the theory of mean-field spin glasses. The new techniques are
broadly applicable, and lead to novel algorithmic and statistical consequences.


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.
<!-- ==December 6, TBA ==-->
 
== 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.
 
== Thursday, October 2, [http://www.math.wisc.edu/~jyin/jun-yin.html Jun Yin], [http://www.math.wisc.edu/ UW-Madison]  ==
 
Title: '''Anisotropic local laws for random matrices'''
 
Abstract:
In this talk, we introduce a new method of deriving  local laws of random matrices.  As 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]  ==
 
No seminar due to [http://www.math.northwestern.edu/mwp/ Midwest Probability Colloquium].
 
 
== Thursday, October 16, [http://www.math.utah.edu/~firas/ Firas Rassoul-Agha], [http://www.math.utah.edu/ University of Utah]==
 
Title: '''The growth model: Busemann functions, shape, geodesics, and other stories'''
 
Abstract:
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 interface.  This is joint work with Nicos Georgiou and Timo Seppalainen.
 
 
== Thursday, November 6, Vadim Gorin, [http://www-math.mit.edu/people/profile.php?pid=1415 MIT]  ==
 
Title: '''Multilevel Dyson Brownian Motion and its edge limits.'''
 
Abstract: The GUE Tracy-Widom distribution is known to govern the large-time asymptotics for a variety of
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.)
 
==<span style="color:red"> Friday</span>, November 7, [http://tchumley.public.iastate.edu/ Tim Chumley], [http://www.math.iastate.edu/ Iowa State University] ==
 
<span style="color:darkgreen">Please note the unusual day.</span>
 
Title: '''Random billiards and diffusion'''
 
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.
 
== Thursday, November 13, [http://www.math.wisc.edu/~seppalai/ Timo Seppäläinen], [http://www.math.wisc.edu/ UW-Madison]==
 
Title: '''Variational formulas for directed polymer and percolation models'''
 
Abstract:
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.
 
 
 
== <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>==
 
<span style="color:darkgreen">Please note the unusual time and room.</span>
 
Title: '''Some phase transitions in the stochastic block model'''
 
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.
 
== Thursday, December 4, Arjun Krishnan, [http://www.fields.utoronto.ca/ Fields Institute] ==
 
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]]

Revision as of 15:51, 29 November 2018


Fall 2018

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 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


Friday, August 10, 10am, B239 Van Vleck András Mészáros, Central European University, Budapest

Title: The distribution of sandpile groups of random regular graphs

Abstract: We study the distribution of the sandpile group of random [math]\displaystyle{ d }[/math]-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the [math]\displaystyle{ p }[/math]-Sylow subgroup of the sandpile group is a given [math]\displaystyle{ p }[/math]-group [math]\displaystyle{ P }[/math], is proportional to [math]\displaystyle{ |\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.

Our results extends a recent theorem of Huang saying that the adjacency matrices of random [math]\displaystyle{ d }[/math]-regular directed graphs are invertible with high probability to the undirected case.


September 20, Hao Shen, UW-Madison

Title: Stochastic quantization of Yang-Mills

Abstract: "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. 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].


September 27, Timo Seppäläinen UW-Madison

Title:Random walk in random environment and the Kardar-Parisi-Zhang class

Abstract:This talk concerns a relationship between two much-studied classes of models of motion in a random medium, namely random walk in random environment (RWRE) and the Kardar-Parisi-Zhang (KPZ) universality class. Barraquand and Corwin (Columbia) discovered that in 1+1 dimensional RWRE in a dynamical beta environment the correction to the quenched large deviation principle obeys KPZ behavior. In this talk we condition the beta walk to escape at an atypical velocity and show that the resulting Doob-transformed RWRE obeys the KPZ wandering exponent 2/3. Based on joint work with Márton Balázs (Bristol) and Firas Rassoul-Agha (Utah).

October 4, Elliot Paquette, OSU

Title: Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble

Abstract: The characteristic polynomial of the Gaussian beta--ensemble can be represented, via its tridiagonal model, as an entry in a product of independent random two--by--two matrices. For a point z in the complex plane, at which the transfer matrix is to be evaluated, this product of transfer matrices splits into three independent factors, each of which can be understood as a different dynamical system in the complex plane. Conjecturally, we show that the characteristic polynomial is always represented as product of at most three terms, an exponential of a Gaussian field, the stochastic Airy function, and a diffusion similar to the stochastic sine equation. We explain the origins of this decomposition, and we show partial progress in establishing part of it.

Joint work with Diane Holcomb and Gaultier Lambert.

October 11, Chris Janjigian, University of Utah

Title: Busemann functions and Gibbs measures in directed polymer models on Z^2

Abstract: We consider the model of a nearest-neighbor random walk on the planar square lattice in a general iid space-time potential, which is also known as a directed polymer in a random environment. We prove results on existence, uniqueness (and non-uniqueness), and the law of large numbers for semi-infinite path measures. Our main tools are the Busemann functions, which are families of stochastic processes obtained through limits of ratios of partition functions.

Based on joint work with Firas Rassoul-Agha

October 18-20, Midwest Probability Colloquium, No Seminar

October 25, Promit Ghosal, Columbia

Title: Tails of the KPZ equation

Abstract: The KPZ equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. It is related to another important SPDE, namely, the stochastic heat equation (SHE). In this talk, we focus on the tail probabilities of the solution of the KPZ equation. For instance, we investigate the probability of the solution being smaller or larger than the expected value. Our analysis is based on an exact identity between the KPZ equation and the Airy point process (which arises at the edge of the spectrum of the random Hermitian matrices) and the Brownian Gibbs property of the KPZ line ensemble.

This talk will be based on a joint work with my advisor Prof. Ivan Corwin.

November 1, James Melbourne, University of Minnesota

Title: Upper bounds on the density of independent vectors under certain linear mappings

Abstract: Using functional analytic techniques and rearrangement, we prove anti-concentration results for the linear images of independent random variables, in the form of density upper bounds. For continuous variables the results unify and sharpen Bobkov-Chistyakov's for independent sums of vectors and Rudelson-Vershynin's bounds on projections of independent coordinates. For integer valued variables the techniques reduce finding the maximum of the probability mass function of a sum of independent variables, to the case that each variable is uniform on a contiguous interval. This problem is approached through analysis of characteristic functions and new $L^p$ bounds on the Dirichlet and Fejer Kernel are obtained and used to derive a discrete analog of Bobkov-Chistyakov.

November 8, Thomas Leblé, NYU

Title: The Sine-beta process: DLR equations and applications

Abstract: One-dimensional log-gases, or Beta-ensembles, are statistical physics models finding an incarnation in random matrix theory. Their limit behavior at microscopic scale is known as the Sine-beta process, its original description involves systems of coupled SDE's. In a joint work with D. Dereudre, A. Hardy, and M. Maïda, we give a new description of Sine-beta as an "infinite volume Gibbs measure", using the Dobrushin-Lanford-Ruelle (DLR) formalism, and we use it to prove the rigidity of the process, in the sense of Ghosh-Peres. Another application is a CLT for fluctuations of linear statistics.


November 22, Thanksgiving Break, No Seminar

Monday, November 26, 4pm, Van Vleck 911 Vadim Gorin, MIT

  Please note the unusual day, time,
  and room.

Title: Macroscopic fluctuations through Schur generating functions

Abstract: I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.

It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.


Wednesday, December 5 at 4pm in Van Vleck 911 Subhabrata Sen, MIT and Microsoft Research New England

  Please note the unusual day, time,
  and room.


Title: Random graphs, Optimization, and Spin glasses

Abstract: Combinatorial optimization problems are ubiquitous in diverse mathematical applications. The desire to understand their “typical” behavior motivates a study of these problems on random instances. In spite of a long and rich history, many natural questions in this domain are still intractable to rigorous mathematical analysis. Graph cut problems such as Max-Cut and Min-bisection are canonical examples in this class. On the other hand, physicists study these questions using the non-rigorous “replica” and “cavity” methods, and predict complex, intriguing features. In this talk, I will describe some recent progress in our understanding of their typical properties on random graphs, obtained via connections to the theory of mean-field spin glasses. The new techniques are broadly applicable, and lead to novel algorithmic and statistical consequences.


Past Seminars