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Probability Seminar

Fall 2017

Date:September 18, 2017
Location:BLOC 220
Speaker:Boris Hanin, TAMU
Title:Local Universality for Random Waves on Riemannian Manifolds
Abstract:Random waves on a Riemannian manifold are a Gaussian model for eigenfunctions of the Laplacian. This model first arose in Berry's random wave conjecture from the 1970's, which states that random waves are good semiclassical models for deterministic wavefunctions in chaotic quantum systems. I will explain what this means and give some examples of why this conjecture is so far from being proved. I will then talk about some ongoing work with Yaiza Canzani about local universality for random waves. The idea is that, just like for random matrix models, random waves have universal scaling limits under some generic assumptions. This local universality allows one to say quite a bit about the size and topology (e.g. number of connected components) of zero sets of random waves. I will state some of these results and a few open questions as well.

Date:September 25, 2017
Location:BLOC 220
Speaker:Gilles Pisier, TAMU & Paris 6
Title:Subgaussian random variables and generalizations

Date:October 9, 2017
Location:BLOC 220
Speaker:Eviatar Procaccia and Yuan Zhang, TAMU
Title:On covering monotonic paths with simple random walk
Abstract:We study the probability that a $d$ dimensional simple random walk (or the first $L$ steps of it) covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball. We show that among all such paths, the one that maximizes the covering probability is the monotonic increasing one that stays within distance 1 from the diagonal. As a result, we can obtain an exponential upper bound on the decaying rate of covering probability of any such path when d≥4 and a $\log$ correction for $d=3$. Interesting conjectures and open questions will be presented. The talk will be split between Eviatar and Yuan. Eviatar will present the background and Yuan the combinatorics part.

Date:November 3, 2017
Location:BLOC 220
Title:UT Austin - TAMU probability day
Abstract: Please register by October 25th.

Date:November 20, 2017
Location:BLOC 220
Speaker:Jiayan Ye, TAMU
Title:Continuity of cheeger constant in super-critical percolation
Abstract:Abstract: We consider the super-critical bond percolation on $Z^d$ with $d \geq 3$ and $p > p_c(Z^d)$. In particular, we study the subgraphs of $C_{\infty} \cap [-n, n]^d$ with minimal cheeger constant, where $C_{\infty}$ is the unique infinite open cluster on $Z^d$. Recently, Gold proved that the subgraphs converge to a deterministic shape almost surely. We prove that this deterministic shape is Hausdrorff - continuous in the percolation parameter $p$. This is joint work with Eviatar Procaccia.