# 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)
TBA

Jiaming Xia, UPenn

TBA

• Tuesday September 28, 2021 at 15:30, Penn (David Rittenhouse Lab A-1)
TBA

Minjae Park, MIT

TBA