# Probability Seminar

# 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) via zoom https://uwmadison.zoom.us/j/99288619661

**Percolation Exponent, Conformal Radius for SLE, and Liouville Structure Constant**

In recent years, a technique has been developed to compute the conformal radii of random domains defined by SLE curves, which is based on the coupling between SLE and Liouville quantum gravity (LQG). Compared to prior methods that compute SLE related quantities via its coupling with LQG, the crucial new input is the exact solvability of structure constants in Liouville conformal field theory. It appears that various percolation exponents can be expressed in terms of conformal radii that can be computed this way. This includes known exponents such as the one-arm and polychromatic

two-arm exponents, as well as the backbone exponents, which is unknown previously. In this talk we will review this method using the derivation of the backbone exponent as an example, based on a joint work with Nolin, Qian, and Sun.

## April 11, 2024: Bjoern Bringman (Princeton)

**Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions.**

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton.

## April 18, 2024: Christopher Janjigian (Purdue)

**Infinite geodesics and Busemann functions in inhomogeneous exponential last passage percolation**

This talk will discuss some recent progress on understanding the structure of semi-infinite geodesics and their associated Busemann functions in the inhomogeneous exactly solvable exponential last-passage percolation model. In contrast to the homogeneous model, this generalization admits linear segments of the limit shape and an associated richer structure of semi-infinite geodesic behaviors. Depending on certain choices of the inhomogeneity parameters, we show that the model exhibits new behaviors of semi-infinite geodesics, which include wandering semi-infinite geodesics with no asymptotic direction, isolated asymptotic directions of semi-infinite geodesics, and non-trivial intervals of directions with no semi-infinite geodesics.

Based on joint work-in-progress with Elnur Emrah (Bristol) and Timo Seppäläinen (Madison)

## April 25, 2024: Colin McSwiggen (NYU)

**Large deviations and multivariable special functions**

This talk introduces techniques for using the large deviations of interacting particle systems to study the large-N asymptotics of generalized Bessel functions. These functions arise from a versatile approach to special functions known as Dunkl theory, and they include as special cases most of the spherical integrals that have captured the attention of random matrix theorists for more than two decades. I will give a brief introduction to Dunkl theory and then present a result on the large-N limits of generalized Bessel functions, which unifies several results on spherical integrals in the random matrix theory literature. These limits follow from a large deviations principle for radial Dunkl processes, which are generalizations of Dyson Brownian motion. If time allows, I will discuss some further results on large deviations of radial Heckman-Opdam processes and/or applications to asymptotic representation theory. Joint work with Jiaoyang Huang.

## May 2, 2024: Anya Katsevich (MIT)

**The Laplace approximation in high-dimensional Bayesian inference**

Computing integrals against a high-dimensional posterior is the major computational bottleneck in Bayesian inference. A popular technique to reduce this computational burden is to use the Laplace approximation, a Gaussian distribution, in place of the true posterior. Despite its widespread use, the Laplace approximation's accuracy in high dimensions is not well understood. The body of existing results does not form a cohesive theory, leaving open important questions e.g. on the dimension dependence of the approximation rate. We address many of these questions through the unified framework of a new, leading order asymptotic decomposition of high-dimensional Laplace integrals. In particular, we (1) determine the tight dimension dependence of the approximation error, leading to the tightest known Bernstein von Mises result on the asymptotic normality of the posterior, and (2) derive a simple correction to this Gaussian distribution to obtain a higher-order accurate approximation to the posterior.