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[[Probability | Back to Probability Group]]


= Spring 2021 =
[[Past Seminars]]


<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.
= Spring 2024 =
<b>We  usually end for questions at 3:20 PM.</b>
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]
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 join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
== January 25, 2024: Tatyana Shcherbina (UW-Madison) ==
'''Characteristic polynomials of sparse non-Hermitian random matrices'''
== January 28, 2021, no seminar  ==


== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==
We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of sparse non-Hermitian random matrices $X_n$ whose entries have the form $x_{jk}=d_{jk}w_{jk}$ with iid complex standard Gaussian $w_{jk}$ and normalized iid Bernoulli$(p)$ $d_{jk}$.  If $p\to\infty$, the local asymptotic behavior of the second correlation function of characteristic polynomials near $z_0\in \mathbb{C}$ coincides with those for  Ginibre ensemble of non-Hermitian matrices with iid Gaussian entries: it converges to a determinant of the Ginibre kernel in the bulk $|z_0|<1$, and it is factorized if $|z_0|>1$. It appears, however, that for the finite $p>0$, the behavior is different and it exhibits the transition between three different regimes depending on values $p$ and $|z_0|^2$.  This is the joint work with Ie. Afanasiev.  


'''Dynamic polymers: invariant measures and ordering by noise'''
== February 1, 2024: [https://lopat.to/index.html Patrick Lopatto (Brown)] ==
'''Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices'''


We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force-One Solution principle holds.
We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.
== February 8, 2024: Benoit Dagallier (NYU), online talk: https://uwmadison.zoom.us/j/95724628357 ==
'''Stochastic dynamics and the Polchinski equation'''


== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford)  ==
I will discuss a general framework to obtain large scale information in statistical mechanics and field theory models. The basic, well known idea is to build a dynamics that samples from the model and control its long time behaviour. There are many ways to build such a dynamics, the Langevin dynamics being a typical example. In this talk I will introduce another, the Polchinski dynamics, based on renormalisation group ideas. The dynamics is parametrised by a parameter representing a certain notion of scale in the model under consideration. The Polchinski dynamics has a number of interesting properties that make it well suited to study large-dimensional models. It is also known under the name stochastic localisation. I will mention a number of recent applications of this dynamics, in particular to prove functional inequalities via a generalisation of Bakry and Emery's convexity-based argument. The talk is based on joint work with Roland Bauerschmidt and Thierry Bodineau and the recent review paper <nowiki>https://arxiv.org/abs/2307.07619</nowiki> .


'''Non-stationary fluctuations for some non-integrable models'''
== February 15, 2024: [https://math.temple.edu/~tue86896/ Brian Rider (Temple)] ==
'''A matrix model for conditioned Stochastic Airy'''


We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.
There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure.  What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. Here I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level.  I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).


== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==
== February 22, 2024: No talk this week ==
'''TBA'''


'''Signature moments to characterize laws of stochastic processes'''
== February 29, 2024:  Zongrui Yang (Columbia) ==
'''Stationary measures for integrable models with two open boundaries'''


The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.
We present two methods to study the stationary measures of integrable systems with two open boundaries. The first method is based on Askey-Wilson signed measures, which is illustrated for the open asymmetric simple exclusion process and the six-vertex model on a strip. The second method is based on two-layer Gibbs measures and is illustrated for the geometric last-passage percolation and log-gamma polymer on a strip. This talk is based on joint works with Yizao Wang, Jacek Wesolowski, Guillaume Barraquand and Ivan Corwin.


== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==
== March 7, 2024: Atilla Yilmaz (Temple) ==
'''Stochastic homogenization of nonconvex Hamilton-Jacobi equations'''


'''Random matrices, random groups, singular values, and symmetric functions'''
After giving a self-contained introduction to the qualitative homogenization of Hamilton-Jacobi (HJ) equations in stationary ergodic media in spatial dimension ''d ≥ 1'', I will focus on the case where the Hamiltonian is nonconvex, and highlight some interesting differences between: (i) periodic vs. truly random media; (ii) ''d = 1'' vs. ''d ≥ 2''; and (iii) inviscid vs. viscous HJ equations.


Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.
== March 14, 2024: Eric Foxall (UBC Okanagan) ==
'''Some uses of ordered representations in finite-population exchangeable ancestry models''' (ArXiv: https://arxiv.org/abs/2104.00193)


== March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==
For a population model that encodes parent-child relations, an ordered representation is a partial or complete labelling of individuals, in order of their descendants’ long-term success in some sense, with respect to which the ancestral structure is more tractable. The two most common types are the lookdown and the spinal decomposition(s), used respectively to study exchangeable models and Markov branching processes. We study the lookdown for an exchangeable model with a fixed, arbitrary sequence of natural numbers, describing population size over time. We give a simple and intuitive construction of the lookdown via the complementary notions of forward and backward neutrality. We discuss its connection to the spinal decomposition in the setting of Galton-Watson trees. We then use the lookdown to give sufficient conditions on the population sequence for the existence of a unique infinite line of descent. For a related but slightly weaker property, takeover, the necessary and sufficient conditions are more easily expressed: infinite time passes on the coalescent time scale. The latter property is also related to the following question of identifiability: under what conditions can some or all of the lookdown labelling be determined by the unlabelled lineages? A reasonably good answer can be obtained by comparing extinction times and relative sizes of lineages.


'''The Coleman correspondence at the free fermion point'''
== March 21, 2024: Semon Rezchikov (Princeton) ==
'''Renormalization, Diffusion Models, and Optimal Transport'''


Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization.  
To this end, we will explain how Polchinski’s formulation of the renormalization group of a statistical field theory can be seen as a gradient flow equation for a relative entropy functional. We will review some related work applying this idea to problems in mathematical physics; subsequently, we will explain how this idea can be used to design adaptive bridge sampling schemes for lattice field theories based on diffusion models which learn the RG flow of the theory.  Based on joint work with Jordan Cotler.
I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions.
I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions.
This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent.
We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$.  
This is joint work with C. Webb (arXiv:2010.07096).


== March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester)  ==
== March 28, 2024: Spring Break ==
'''TBA'''


'''The limit shape of the Leaky Abelian Sandpile Model'''
== April 4, 2024: Zijie Zhuang (Upenn)  online talk ==
'''TBA'''


The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.
== April 11, 2024: Bjoern Bringman (Princeton) ==
'''TBA'''


We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.
== April 18, 2024:  Christopher Janjigian (Purdue) ==
'''TBA'''


We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.
== April 25, 2024: Colin McSwiggen (NYU) ==
'''TBA'''


== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==
== May 2, 2024: Anya Katsevich (MIT) ==
 
'''TBA'''
'''On the joint moments of characteristic polynomials of random unitary matrices'''
I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.
 
== March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==
'''Fluctuations of particle density  for open ASEP'''
 
I will review results on fluctuations of particle density for the open Asymmetric Simple Exclusion Process. I will explain the statements and the Laplace transform duality arguments that appear in the proofs.
 
The talk is based on past and ongoing projects with  Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.
 
== April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University)  ==
'''Motion by mean curvature in interacting particle systems'''
 
There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term.  These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et. al there were two nontrivial stationary distributions.
 
 
== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==
 
== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics)  ==
 
== April 16, 2021, [http://www.mathjunge.com/ Matthew Junge] (CUNY) <span style="color:red">FRIDAY at 2:25pm, joint with</span> [https://www.math.wisc.edu/wiki/index.php/Applied/ACMS ACMS]  ==
 
== April 22, 2021, [https://www.maths.ox.ac.uk/people/benjamin.fehrman Benjamin Fehrman] (Oxford)  ==
'''Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE'''
 
Abstract:  Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning.  We will focus, in particular, on the zero range process and the symmetric simple exclusion process.  The large-scale behavior of these systems is essentially deterministic, and is described in terms of a hydrodynamic limit.  However, the particle process does exhibit large fluctuations away from its mean.  Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate.
 
In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process.  The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise.  The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.
 
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford)  ==
 
[[Past Seminars]]

Latest revision as of 14:53, 18 March 2024

Back to Probability Group

Past Seminars

Spring 2024

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

January 25, 2024: Tatyana Shcherbina (UW-Madison)

Characteristic polynomials of sparse non-Hermitian random matrices

We consider the asymptotic local behavior of the second correlation functions of the characteristic polynomials of sparse non-Hermitian random matrices $X_n$ whose entries have the form $x_{jk}=d_{jk}w_{jk}$ with iid complex standard Gaussian $w_{jk}$ and normalized iid Bernoulli$(p)$ $d_{jk}$.  If $p\to\infty$, the local asymptotic behavior of the second correlation function of characteristic polynomials near $z_0\in \mathbb{C}$ coincides with those for  Ginibre ensemble of non-Hermitian matrices with iid Gaussian entries: it converges to a determinant of the Ginibre kernel in the bulk $|z_0|<1$, and it is factorized if $|z_0|>1$. It appears, however, that for the finite $p>0$, the behavior is different and it exhibits the transition between three different regimes depending on values $p$ and $|z_0|^2$.  This is the joint work with Ie. Afanasiev.  

February 1, 2024: Patrick Lopatto (Brown)

Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov–Hiary–Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.

February 8, 2024: Benoit Dagallier (NYU), online talk: https://uwmadison.zoom.us/j/95724628357

Stochastic dynamics and the Polchinski equation

I will discuss a general framework to obtain large scale information in statistical mechanics and field theory models. The basic, well known idea is to build a dynamics that samples from the model and control its long time behaviour. There are many ways to build such a dynamics, the Langevin dynamics being a typical example. In this talk I will introduce another, the Polchinski dynamics, based on renormalisation group ideas. The dynamics is parametrised by a parameter representing a certain notion of scale in the model under consideration. The Polchinski dynamics has a number of interesting properties that make it well suited to study large-dimensional models. It is also known under the name stochastic localisation. I will mention a number of recent applications of this dynamics, in particular to prove functional inequalities via a generalisation of Bakry and Emery's convexity-based argument. The talk is based on joint work with Roland Bauerschmidt and Thierry Bodineau and the recent review paper https://arxiv.org/abs/2307.07619 .

February 15, 2024: Brian Rider (Temple)

A matrix model for conditioned Stochastic Airy

There are three basic flavors of local limit theorems in random matrix theory, connected to the spectral bulk and the so-called soft and hard edges. There also abound a collection of more exotic limits which arise in models that posses degenerate (or “non-regular”) points in their equilibrium measure.  What is more, there is typically a natural double scaling about these non-regular points, producing limit laws that transition between the more familiar basic flavors. Here I will describe a general beta matrix model for which the appropriate double scaling limit is the Stochastic Airy Operator, conditioned on having no eigenvalues below a fixed level.  I know of no other random matrix double scaling fully characterized outside of beta = 2. This is work in progress with J. Ramirez (University of Costa Rica).

February 22, 2024: No talk this week

TBA

February 29, 2024: Zongrui Yang (Columbia)

Stationary measures for integrable models with two open boundaries

We present two methods to study the stationary measures of integrable systems with two open boundaries. The first method is based on Askey-Wilson signed measures, which is illustrated for the open asymmetric simple exclusion process and the six-vertex model on a strip. The second method is based on two-layer Gibbs measures and is illustrated for the geometric last-passage percolation and log-gamma polymer on a strip. This talk is based on joint works with Yizao Wang, Jacek Wesolowski, Guillaume Barraquand and Ivan Corwin.

March 7, 2024: Atilla Yilmaz (Temple)

Stochastic homogenization of nonconvex Hamilton-Jacobi equations

After giving a self-contained introduction to the qualitative homogenization of Hamilton-Jacobi (HJ) equations in stationary ergodic media in spatial dimension d ≥ 1, I will focus on the case where the Hamiltonian is nonconvex, and highlight some interesting differences between: (i) periodic vs. truly random media; (ii) d = 1 vs. d ≥ 2; and (iii) inviscid vs. viscous HJ equations.

March 14, 2024: Eric Foxall (UBC Okanagan)

Some uses of ordered representations in finite-population exchangeable ancestry models (ArXiv: https://arxiv.org/abs/2104.00193)

For a population model that encodes parent-child relations, an ordered representation is a partial or complete labelling of individuals, in order of their descendants’ long-term success in some sense, with respect to which the ancestral structure is more tractable. The two most common types are the lookdown and the spinal decomposition(s), used respectively to study exchangeable models and Markov branching processes. We study the lookdown for an exchangeable model with a fixed, arbitrary sequence of natural numbers, describing population size over time. We give a simple and intuitive construction of the lookdown via the complementary notions of forward and backward neutrality. We discuss its connection to the spinal decomposition in the setting of Galton-Watson trees. We then use the lookdown to give sufficient conditions on the population sequence for the existence of a unique infinite line of descent. For a related but slightly weaker property, takeover, the necessary and sufficient conditions are more easily expressed: infinite time passes on the coalescent time scale. The latter property is also related to the following question of identifiability: under what conditions can some or all of the lookdown labelling be determined by the unlabelled lineages? A reasonably good answer can be obtained by comparing extinction times and relative sizes of lineages.

March 21, 2024: Semon Rezchikov (Princeton)

Renormalization, Diffusion Models, and Optimal Transport

To this end, we will explain how Polchinski’s formulation of the renormalization group of a statistical field theory can be seen as a gradient flow equation for a relative entropy functional. We will review some related work applying this idea to problems in mathematical physics; subsequently, we will explain how this idea can be used to design adaptive bridge sampling schemes for lattice field theories based on diffusion models which learn the RG flow of the theory.  Based on joint work with Jordan Cotler.

March 28, 2024: Spring Break

TBA

April 4, 2024: Zijie Zhuang (Upenn) online talk

TBA

April 11, 2024: Bjoern Bringman (Princeton)

TBA

April 18, 2024: Christopher Janjigian (Purdue)

TBA

April 25, 2024: Colin McSwiggen (NYU)

TBA

May 2, 2024: Anya Katsevich (MIT)

TBA