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[[Probability | Back to Probability Group]]


= Fall 2020 =
[[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 Fall 2020 the seminar is being run online.
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'''
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==


'''Pre-Talk:'''
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.  


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


Abstract: Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.
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'''


'''Talk:'''
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> .


Title: '''Effective Theory of Deep Neural Networks'''  
== February 15, 2024: [https://math.temple.edu/~tue86896/ Brian Rider (Temple)] ==
'''A matrix model for conditioned Stochastic Airy'''


Abstract: Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.
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 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin)  ==
== February 22, 2024: No talk this week ==
'''TBA'''


== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen], [https://mscs.uic.edu/ UIC] ==
== February 29, 2024: Zongrui Yang (Columbia) ==
'''Stationary measures for integrable models with two open boundaries'''


Title: '''Roots of random polynomials near the unit circle'''
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.


Abstract: It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle.  Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei.  We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle.  Based on joint work with Julian Sahasrabudhe.
== March 7, 2024: Atilla Yilmaz (Temple) ==
'''Stochastic homogenization of nonconvex Hamilton-Jacobi equations'''


== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen], [https://statistics.fas.harvard.edu/ Harvard] ==
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.


Title: '''TBA'''
== 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)


Abstract: TBA
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.


== November 12, 2020, [http://stanford.edu/~ajdunl2/ Alexander Dunlap], [https://cims.nyu.edu/ NYU Courant Institute] ==
== March 21, 2024: Semon Rezchikov (Princeton) ==
'''Renormalization, Diffusion Models, and Optimal Transport'''


Title: '''TBA'''
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.


Abstract: TBA
== March 28, 2024: Spring Break ==
'''TBA'''


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


[[Past Seminars]]
== 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'''

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