Difference between revisions of "Probability Seminar"

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(Friday, November 17, 2017, 1pm, Van Vleck B223, Karl Leichty DePaul University)
(Wednesday, December 5 at 4pm in Van Vleck 911 Subhabrata Sen, MIT and Microsoft Research New England)
 
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__NOTOC__
 
__NOTOC__
  
= Fall 2017 =
+
= 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>We  usually end for questions at 3:15 PM.</b>
  
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.
+
If you would like to sign up for the email list to receive seminar announcements then please send an email to  
 +
[mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu]
  
  
== Thursday, September 14, 2017, [https://math.temple.edu/~brider/ Brian Rider] [https://math.temple.edu/ Temple University] ==
 
  
'''A universality result for the random matrix hard edge'''
+
==<span style="color:red"> Friday, August 10, 10am, B239 Van Vleck </span> András Mészáros, Central European University, Budapest ==
  
The hard edge refers to the distribution of the smallest singular value for certain ensembles of random matrices, or, and what is the same, that of the minimal point of a logarithmic gas constrained to the positive half line. For any "inverse temperature" and “quadratic" potential the possible limit laws (as the dimension, or number of particles, tends to infinity) was characterized by Jose Ramirez and myself in terms of the spectrum of a (random) diffusion generator. Here we show this picture persists for more general convex polynomial potentials. Joint work with Patrick Waters.
 
  
<!-- == Thursday, September 21, 2017, TBA==-->
+
Title: '''The distribution of sandpile groups of random regular graphs'''
  
<!-- == Thursday, September 28, 2017, TBA ==
+
Abstract:
== Thursday, October 5, 2017 ==
+
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.
== Thursday, October 12, 2017 == -->
+
== Thursday, October 19, 2017  [https://sites.google.com/wisc.edu/vjog/ Varun Jog], [https://www.engr.wisc.edu/department/electrical-computer-engineering/ UW-Madison ECE] and [https://graingerinstitute.engr.wisc.edu/ Grainger Institute] ==
+
  
Title: '''Teaching and learning in uncertainty'''
+
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.
  
Abstract:
+
<!-- ==September 13, TBA == -->
We investigate a simple model for social learning with two characters: a teacher and a student. The teacher's goal is to teach the student the state of the world <math>\Theta</math>, however, the teacher herself is not certain about <math>\Theta</math> and needs to simultaneously learn it and teach it. We examine several natural strategies the teacher may employ to make the student learn as fast as possible. Our primary technical contribution is analyzing the exact learning rates for these strategies by studying the large deviation properties of the sign of a transient random walk on <math>\mathbb Z</math>.
+
  
== Thursday, October 26, 2017, [http://www.math.toronto.edu/matetski/ Konstantin Matetski] [https://www.math.toronto.edu/ Toronto] ==
+
==September 20, [http://math.columbia.edu/~hshen/ Hao Shen], [https://www.math.wisc.edu/ UW-Madison] ==
  
Title: '''The KPZ fixed point'''
+
Title: '''Stochastic quantization of Yang-Mills'''
  
 
Abstract:
 
Abstract:
The KPZ fixed point is the Markov process at the centre of the KPZ universality class. In the talk we describe the exact solution of the totally asymmetric simple exclusion process, which is one of the models in the KPZ universality class, and provide a description of the KPZ fixed point in the 1:2:3 scaling limit. This is a joint work with Jeremy Quastel and Daniel Remenik.
+
"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, November 2, 2017, TBA ==-->
 
  
== Thursday, November 9, 2017, Chen Jia, University of Texas at Dallas  ==
 
  
 +
==September 27, [https://www.math.wisc.edu/~seppalai/  Timo Seppäläinen] [https://www.math.wisc.edu/ UW-Madison] ==
  
'''Mathematical foundation of nonequilibrium fluctuation-dissipation theorems and a biological application'''
+
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).
  
The fluctuation-dissipation theorem (FDT) for equilibrium states is one of the classical results in equilibrium statistical physics. In recent years, many efforts have been devoted to generalizing the classical FDT to systems far from equilibrium. This was considered as one of the most significant progress of nonequilibrium statistical physics over the past two decades. In this talk, I will introduce our recent work on the rigorous mathematical foundation of the nonequilibrium FDTs for inhomogeneous diffusion processes and inhomogeneous continuous-time Markov chains. I will also talk about the application of the nonequilibrium FDTs to a practical biological problem called sensory adaptation.
+
==October 4, [https://people.math.osu.edu/paquette.30/  Elliot Paquette], [https://math.osu.edu/ OSU] ==
  
== Thursday, November 16, 2017, [http://louisfan.web.unc.edu/ Louis Fan], [http://www.math.wisc.edu/ UW-Madison]  ==
+
Title: '''Distributional approximation of the characteristic polynomial of a Gaussian beta-ensemble'''
  
Title: '''Stochastic and deterministic spatial models for complex systems'''
+
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.
  
Abstract:
+
Joint work with Diane Holcomb and Gaultier Lambert.
  
Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.
+
==October 11, [https://www.math.utah.edu/~janjigia/ Chris Janjigian], [https://www.math.utah.edu/ University of Utah] ==
  
In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.
 
  
== <span style="color:red"> Friday,</span> November 17, 2017,  <span style="color:red"> 1pm, Van Vleck B223, </span> [http://math.depaul.edu/kliechty/ Karl Leichty] [https://csh.depaul.edu/academics/mathematical-sciences/Pages/default.aspx DePaul University] ==
+
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.
  
<div style="width:350px;height:50px;border:5px solid black">
+
Based on joint work with Firas Rassoul-Agha
<b><span style="color:red"> Please note the unusual room, day, and time </span></b>
+
 
 +
==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 ==
 +
 
 +
 
 +
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, [https://math.umn.edu/directory/james-melbourne James Melbourne], [https://math.umn.edu/ 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, [https://cims.nyu.edu/~thomasl/ Thomas Leblé], [https://cims.nyu.edu/ 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 15, TBA == -->
 +
 
 +
==November 22, [https://en.wikipedia.org/wiki/Thanksgiving Thanksgiving] Break, No Seminar ==
 +
 
 +
==  <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]  ==
 +
 
 +
 
 +
<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>
 
</div>
  
Title: '''Nonintersecting Brownian motions on the unit circle'''
+
Title: '''Macroscopic fluctuations through Schur generating functions'''
+
 
 
Abstract:
 
Abstract:
+
I will talk about a special class of large-dimensional stochastic systems with
Nonintersecting Brownian bridges on the unit circle form a determinantal point process whose kernel is expressed in terms of a system of discrete orthogonal polynomials which may be studied using Riemann--Hilbert techniques. If the Brownian motions have a drift, then the weight of the orthogonal polynomials becomes complex. I will discuss the tacnode and k-tacnode processes, which are related to the Painleve II function, as scaling limits of Nonintersecting Brownian motions on the unit circle and will discuss some of the features and difficulties of Riemann--Hilbert analysis of discrete orthogonal polynomials with varying complex weights.
+
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.
This is joint work with Dong Wang and Robert Buckingham.
+
  
== Thursday, November 30, 2017, TBA ==
+
It is believed that macroscopic fluctuations in such systems are universally
== Thursday, December 7, 2017,  TBA ==
+
described by log-correlated Gaussian fields. I will present an approach to
== Thursday, December 14, 2017, TBA ==
+
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] ==
 +
 +
 +
<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: '''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.
  
  
 +
<!-- ==December 6, TBA ==-->
  
 
== ==
 
== ==
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 09:51, 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 d-regular graphs. For the directed model we prove that it follows the Cohen-Lenstra heuristics, that is, the probability that the p-Sylow subgroup of the sandpile group is a given p-group P, is proportional to |\operatorname{Aut}(P)|^{-1}. 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 d-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