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[[Probability | Back to Probability Group]]


= Fall 2021 =
[[Past Seminars]]


<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>  
= Spring 2024 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


We usually end for questions at 3:20 PM.
We usually end for questions at 3:20 PM.


[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
== January 25, 2024: Tatyana Shcherbina (UW-Madison) ==
'''Characteristic polynomials of sparse non-Hermitian random matrices'''


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].
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.  
== September 16, 2021, in person: [https://hanbaeklyu.com/ Hanbaek Lyu] (UW-Madison)  ==


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


'''Scaling limit of soliton statistics of a multicolor box-ball system'''
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'''


The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connection between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of multicolor BBS with random initial configurations. Our analysis uses various methods such as modified Greene-Kleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz.
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> .


== September 23, 2021, no seminar  ==
== February 15, 2024: [https://math.temple.edu/~tue86896/ 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).


== September 30, 2021, in person: [https://mrusskikh.mit.edu/home Marianna Russkikh] (MIT)    ==
== February 22, 2024: No talk this week ==
'''TBA'''


'''Lozenge tilings and the Gaussian free field on a cylinder'''
== February 29, 2024:  Zongrui Yang (Columbia) ==
'''Stationary measures for integrable models with two open boundaries'''


We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.
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.


== October 7, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://n.ethz.ch/~bdembin/home.html Barbara Dembin] (ETH Zurich)   ==
== March 7, 2024: Atilla Yilmaz (Temple) ==
'''Stochastic homogenization of nonconvex Hamilton-Jacobi equations'''


'''The time constant for Bernoulli percolation is Lipschitz continuous strictly above $p_c$'''
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.


We consider the standard model of i.i.d. first passage percolation on $\mathbb Z^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty))>p_c(d)$, it is known that the time constant $\mu_G$ exists. We are interested in the regularity properties of the map $G\mapsto\mu_G$. We study the specific case of distributions of the form $G_p=p\delta_1+(1-p)\delta_\infty$ for $p>p_c(d)$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter $p$. We prove that the function $p\mapsto \mu_{G_p}$ is Lipschitz continuous on every interval $[p_0,1]$, where $p_0>p_c(d)$.
== March 14, 2024: Eric Foxall (UBC Okanagan) ==
This is a joint work with Raphaël Cerf.
'''Some uses of ordered representations in finite-population exchangeable ancestry models''' (ArXiv: https://arxiv.org/abs/2104.00193)


== October 14, 2021, <span style="color:red">UPDATED FORMAT: </span> [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/wisc.edu/evan-sorensen Evan Sorensen] (UW-Madison)  ==
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.


'''Busemann functions and semi-infinite geodesics in a semi-discrete space'''
== March 21, 2024: Semon Rezchikov (Princeton) ==
'''Renormalization, Diffusion Models, and Optimal Transport'''


In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization of  uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that direction. Further, there exists a random set of points, of Hausdorff dimension ½, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and never come back together. We derive these results by studying variational problems for Brownian motion with drift.
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.


== October 21, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [http://stat.columbia.edu/~sumitm/ Sumit Mukherjee] (Columbia)    ==
== March 28, 2024: Spring Break ==
'''TBA'''


'''Fluctuations in Mean Field Ising Models'''
== April 4, 2024: Zijie Zhuang (Upenn)  online talk ==
'''TBA'''


We study fluctuations of the magnetization (average of spins) in an Ising model on a sequence of "well-connected" approximately $d_n$ regular graphs on $n$ vertices. We show that if $d_n\gg n^{1/2}$,  then the fluctuations are universal, and same as that of the Curie–Weiss model, in the entire ferromagnetic parameter regime. We then give a counterexample to show that $d_n\gg n^{1/2}$ is actually tight, in the sense that the limiting distribution changes if $d_n\sim n^{1/2}$ except in the high temperature regime. By refining our argument, we show that in the high temperature regime universality holds for $d_n\gg n^{1/3}$. As a by-product of our proof technique, we prove rates of convergence, as well as exponential concentration for the sum of spins, and tight estimates for several statistics of interest.
== April 11, 2024: Bjoern Bringman (Princeton) ==
'''TBA'''


This is based on joint work with Nabarun Deb at Columbia University.
== April 18, 2024:  Christopher Janjigian (Purdue) ==
'''TBA'''


== October 28, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~wkchen/ Wei-Kuo Chen] (Minnesota)   ==
== April 25, 2024: Colin McSwiggen (NYU) ==
'''TBA'''


'''Grothendieck $L_p$ problem for Gaussian matrices'''
== May 2, 2024: Anya Katsevich (MIT) ==
 
'''TBA'''
The Grothendieck $L_p$ problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit $L_p$ ball. The $p=2$ case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while for $p=\infty$ this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general $p$ and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.
 
== November 4, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://mathematics.stanford.edu/people/mackenzie-simper Mackenzie Simper] (Stanford)  ==
 
'''Double Cosets, Mallows Measure, and a Transvections Markov Chain'''
 
If $G = GL_n(\mathbb{F}_q)$ and $B$ is the subgroup of lower triangular matrices, then the {{math|''B'' \ ''G'' / ''B''}} double cosets are indexed by permutations $S_n$. This is the famous Bruhat decomposition, closely related to the LU decomposition of a matrix. The Markov chain on $G$ generated by random transvections – matrices which fix a hyperplane – induces a Markov chain on $S_n$ with the Mallows measure as stationary distribution. We characterize this process, study the mixing time, and discuss the connection with the number of pivoting steps needed in Gaussian elimination. This is joint work with Persi Diaconis and Arun Ram.
 
== November 11, 2021, in person: [http://web.lfzhang.com/ Lingfu Zhang] (Princeton)    ==
 
== November 18, 2021, in person [http://www.ilt.kharkov.ua/bvi/structure/depart_e/d24/mariya_shcherbina-cv.htm Mariya Shcherbina]: (Kharkov) ==
 
== November 25, 2021, no seminar  ==
 
 
== December 2, 2021, in person: [http://math.uchicago.edu/~xuanw/ Xuan Wu] (Chicago) ==
 
 
== December 9, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.dur.ac.uk/users/sunil.chhita/ Sunil Chhita] (Durham)  ==
 
'''GOE Fluctuations for the maximum of the top path in ASMs'''
 
The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling.  This talk is based on joint work with Arvind Ayyer and Kurt Johansson.
 
 
[[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