# Probability Seminar 2021

The seminar is jointly sponsored by Temple and Penn. The organizers are Brian Rider and Atilla Yilmaz (Temple), and Jian Ding, Robin Pemantle and Xin Sun (Penn).

Talks are Tuesdays 3:30 - 4:30 pm and are held either in Wachman Hall (Temple) or David Rittenhouse Lab (Penn) as indicated below.

For a chronological listing of the talks, click the year above.

• Tuesday September 7, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)
Delocalization and quantum diffusion of random band matrices in high dimensions

Fan Yang, UPenn

We consider a Hermitian random band matrix $H$ on the $d$-dimensional lattice of linear size $L$. Its entries are independent centered complex Gaussian random variables with variances $s_{xy}$, that are negligible if $|x-y|$ exceeds the band width $W$. In dimensions eight or higher, we prove that, as long as $W > L^\epsilon$ for a small constant $\epsilon>0$, with high probability, most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Moreover, we also prove a quantum diffusion result of this model in terms of the Green's function of $H$. Joint work with Horng-Tzer Yau and Jun Yin.

• Tuesday September 14, 2021 at 15:30, Temple (Wachman Hall 617)
Spanning clusters and subcritical connectivity in high-dimensional percolation

Jack Hanson, City College of NY, CUNY

In their study of percolation, physicists have proposed "scaling hypotheses" relating the behavior of the model in the critical ($p = p_c$) and subcritical ($p < p_c$) regimes. We show a version of such a scaling hypothesis for the one-arm probability $\pi(n;p)$ — the probability that the open cluster of the origin has Euclidean diameter at least $n$.

As a consequence of our analysis, we obtain the correct scaling of the lower tail of cluster volumes and the chemical (intrinsic) distances within clusters. We also show that the number of spanning clusters of a side-length $n$ box is tight on scale $n^{d-6}$. A new tool of our analysis is a sharp asymptotic for connectivity probabilities when paths are restricted to lie in half-spaces.

• Tuesday September 21, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)
Hamilton-Jacobi equations for statistical inference problems

Jiaming Xia, UPenn

In this talk, I will first present the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. We compare the limit with the solution to a certain Hamilton-Jacobi equation, following the recent approach by Jean-Christophe Mourrat. The motivation comes from the averaged free energy solving an approximate Hamilton-Jacobi equation. We consider two notions of solutions which are weak solutions and viscosity solutions. The two types of solutions require different treatments and each has its own advantages. At the end of this part, I will show an example of application of our results to a model with i.i.d. entries and symmetric interactions. If time permits, I will talk about the same problem but with a different model, namely, the multi-layer generalized linear model. I will mainly explain the iteration method as an important tool used in our proof. This is based on joint works with Hong-Bin Chen and J.-C. Mourrat, NYU.

• Tuesday September 28, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)
Wilson loop expectations as sums over surfaces in 2D

Minjae Park, MIT

Although lattice Yang-Mills theory on $\mathbb{Z}^d$ is easy to rigorously define, the construction of a satisfactory continuum theory on $\mathbb{R}^d$ is a major open problem when $d\ge 3$. Such a theory should assign a Wilson loop expectation to each suitable collection $\mathcal{L}$ of loops in $\mathbb{R}^d$. One classical approach is to try to represent this expectation as a sum over surfaces with boundary $\mathcal{L}$. There are some formal/heuristic ways to make sense of this notion, but they typically yield an ill-defined difference of infinities.

In this talk, we show how to make sense of Yang-Mills integrals as surface sums for $d=2$, where the continuum theory is already understood. We also obtain an alternative proof of the Makeenko-Migdal equation and a version of the Gross-Taylor expansion. Joint work with Joshua Pfeffer, Scott Sheffield, and Pu Yu.

• Tuesday October 5, 2021 at 15:30, Temple (Wachman Hall 617)
Singularities in the spectrum of random block matrices

David Renfrew, SUNY Binghamton

We consider the density of states of structured Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.

• Tuesday October 12, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)
The local limit theorem on nilpotent groups

Robert Hough, Stony Brook University

Alexopoulos proved local limit theorems for measures with a density and lattice measures in the general setting of groups of moderate growth. On the Heisenberg group, Breuillard's thesis obtained a local limit theorem for general measures subject to a condition on the characteristic function, and asked if this condition can be removed. I will discuss two new local limit theorems, one joint with Diaconis, that treat local limit theorems on nilpotent Lie groups driven by general measures. We prove Breuillard's conjecture and also solve a problem of Diaconis and Saloff-Coste on the mixing of the central coordinate in unipotent matrix walks modulo $p$.﻿

• Tuesday October 19, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)
Integrability of boundary Liouville CFT

Guillaume Remy, Columbia University

Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. Our latest result is derived using conformal welding of random surfaces, in relation with the Schramm-Loewner evolutions. We will also discuss the connection with the conformal blocks of CFT which are fundamental functions determined by conformal invariance that underlie the exact solvability of CFT. Based on joint works with Morris Ang, Promit Ghosal, Xin Sun, Yi Sun and Tunan Zhu.

• Tuesday October 26, 2021 at 15:30, Temple (Wachman Hall 617)
Gaussian, stable, and tempered stable limiting distributions for random walks in cooling random environments

Jonathon Peterson, Purdue University

Random walks in cooling random environments are a model of random walks in dynamic random environments where the random environment is re-sampled at a fixed sequence of times (called the cooling sequence) and the environment remains constant between these re-sampling times. We study the limiting distributions of the walk in the case when distribution on environments is such that a walk in a fixed environment has an $s$-stable limiting distribution for some $s \in (1,2)$. It was previously conjectured that for cooling maps whose gaps between re-sampling times grow polynomially that the model should exhibit a phase transition from Gaussian limits to $s$-stable depending on the exponent of the polynomial growth of the re-sampling gaps. We confirm this conjecture, identifying the precise exponent at which the phase transition occurs and proving that at the critical exponent the limiting distribution is a generalized tempered $s$-stable distribution. The proofs require us to prove some previously unknown facts about one-dimensional random walks in random environments which are of independent interest.