Berkeley Student Probability Seminar, Fall 2026

Deciding order of speakers and topics.

I will give a brief introduction to sphere-homeomorphic random surfaces through four different lenses: the Brownian sphere, the peanosphere, the pure Liouville quantum gravity sphere, and a conformal field theory. The talk will follow Scott Sheffield's survey paper, "What is a random surface?"

In this talk we discuss the results of Michelen and Sahasrabudhe in their paper "Central limit theorems and the geometry of polynomials." In particular, we show that if a discrete random variable $X$ has a generating polynomial which is zero-free near 1, then $X$ is close to a Gaussian in distribution. Time permitting, we may also discuss extensions of this result presented in the same paper, using different zerofreeness properties.

I will discuss two approaches to proving Wilson's algorithm, which generates spanning trees from loop erased random walks on a graph. The first approach is Wilson's original proof, which reduces the problem to a cycle-popping algorithm. The second is Lawler's proof using Green's functions, which also proves Kirchhoff's matrix tree theorem for computing the number of spanning trees of a graph from its Laplacian as a corollary.

The first of two talks on localization schemes, a new framework for analyzing mixing times of Markov chains. I will introduce stochastic localization and its generalization to localization schemes, using Glauber dynamics as a running example. In particular, I will discuss how approximate conservation of variance along the localization process implies a spectral gap.

We continue our discussion of localization schemes from last week, showing how a localization scheme that satisfies approximate conservation of variance can recover the framework of spectral independence. We will then pivot to the related notion of approximate conservation of entropy, which can be used to prove log-Sobolev inequalities for certain chains and thus give optimal mixing time bounds. If time permits, we will survey a few concrete applications of localization schemes to mixing times of Markov chains.