# Probability Seminar

**When**: Thursdays at 2:30 pm**Where**: 901 Van Vleck Hall**Organizers**: Hanbaek Lyu, Tatyana Shcherbyna, David Clancy**To join the probability seminar mailing list:**email probsem+subscribe@g-groups.wisc.edu.**To subscribe seminar lunch announcements:**email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu

# Fall 2024

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

We usually end for questions at 3:20 PM.

## September 5, 2024:

No seminar

## September 12, 2024: Hongchang Ji (UW-Madison)

**Spectral edge of non-Hermitian random matrices**

We report recent progress on spectra of so-called deformed i.i.d. matrices. They are square non-Hermitian random matrices of the form $A+X$ where $X$ has centered i.i.d. entries and $A$ is a deterministic bias, and $A$ and $X$ are on the same scale so that their contributions to the spectrum of $A+X$ are comparable. Under this setting, we present two recent results concerning universal patterns arising in eigenvalue statistics of $A+X$ around its boundary, on macroscopic and microscopic scales. The first result shows that the macroscopic eigenvalue density of $A+X$ typically has a jump discontinuity around the boundary of its support, which is a distinctive feature of $X$ by the \emph{circular law}. The second result is edge universality for deformed non-Hermitian matrices; it shows that the local eigenvalue statistics of $A+X$ around a typical (jump) boundary point is universal, i.e., matches with those of a Ginibre matrix $X$ with i.i.d. standard Gaussian entries.

Based on joint works with A. Campbell, G. Cipolloni, and L. Erd\H{o}s.

## September 19, 2024: Miklos Racz (Northwestern)

**The largest common subtree of uniform attachment trees**

Consider two independent uniform attachment trees with n nodes each -- how large is their largest common subtree? Our main result gives a lower bound of n^{0.83}. We also give some upper bounds and bounds for general random tree growth models. This is based on joint work with Johannes Bäumler, Bas Lodewijks, James Martin, Emil Powierski, and Anirudh Sridhar.

## September 26, 2024: Dmitry Krachun (Princeton)

**A glimpse of universality in critical planar lattice models**

Abstract: Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.

In this talk I will explain some ideas behind the recent result which proves rotational invariance of the FK-percolation model. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz.

Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.

## October 3, 2024: Joshua Cape (UW-Madison)

**A new random matrix: motivation, properties, and applications**

In this talk, we introduce and study a new random matrix whose entries are dependent and discrete valued. This random matrix is motivated by problems in multivariate analysis and nonparametric statistics. We establish its asymptotic properties and provide comparisons to existing results for independent entry random matrix models. We then apply our results to two problems: (i) community detection, and (ii) principal submatrix localization. Based on joint work with Jonquil Z. Liao.

## October 10, 2024: Midwest Probability Colloquium

N/A

## October 17, 2024: Kihoon Seong (Cornell)

**Gaussian fluctuations of focusing Φ^4 measure around the soliton manifold**

I will explain the central limit theorem for the focusing Φ^4 measure in the infinite volume limit. The focusing Φ^4 measure, an invariant Gibbs measure for the nonlinear Schrödinger equation, was first studied by Lebowitz, Rose, and Speer (1988), and later extended by Bourgain (1994), Brydges and Slade (1996), and Carlen, Fröhlich, and Lebowitz (2016).

Rider previously showed that this measure is strongly concentrated around a family of minimizers of the associated Hamiltonian, known as the soliton manifold. In this talk, I will discuss the fluctuations around this soliton manifold. Specifically, we show that the scaled field under the focusing Φ^4 measure converges to white noise in the infinite volume limit, thus identifying the next-order fluctuations, as predicted by Rider.

This talk is based on joint work with Philippe Sosoe (Cornell).

## October 24, 2024: Jacob Richey (Alfred Renyi Institute)

**Stochastic abelian particle systems and self-organized criticality**

Abstract: Activated random walk (ARW) is an 'abelian' particle system that conjecturally exhibits complex behaviors which were first described by physicists in the 1990s, namely self organized criticality and hyperuniformity. I will discuss recent results for ARW and the stochastic sandpile (a related model) on Z and other graphs, plus many open questions.

## October 31, 2024: David Clancy (UW-Madison)

**Likelihood landscape on a known phylogeny**

Abstract: Over time, ancestral populations evolve to become separate species. We can represent this history as a tree with edge lengths where the leaves are the modern-day species. If we know the precise topology of the tree (i.e. the precise evolutionary relationship between all the species), then we can imagine traits (their presence or absence) being passed down according to a symmetric 2-state continuous-time Markov chain. The branch length becomes the probability a parent species has a trait while the child species does not. This length is unknown, but researchers have observed they can get pretty good estimates using maximum likelihood estimation and only the leaf data despite the fact that the number of critical points for the log-likelihood grows exponentially fast in the size of the tree. In this talk, I will discuss why this MLE approach works by showing that the population log-likelihood is strictly concave and smooth in a neighborhood around the true branch length parameters and the size.

This talk is based on joint work with Hanbaek Lyu, Sebastien Roch and Allan Sly.

## November 7, 2024: Zoe Huang (UNC Chapel Hill)

TBD

## November 14, 2024: Deb Nabarun (University of Chicago)

TBD

## November 21, 2024: Reza Gheissari (Northwestern)

TBD

## November 28, 2024: Thanksgiving

No seminar

## December 5, 2024: Erik Bates (NC State)

TBD

# 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.