# Probability Seminar

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)
TBA

Robert Hough, Stony Brook University

TBA

• Tuesday October 19, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)
TBA

Guillaume Remy, Columbia University

TBA

• Tuesday October 26, 2021 at 15:30, Temple (Wachman Hall 617)
TBA

Jonathon Peterson, Purdue University

TBA