# Student Probability Seminar

#### The Mixing Time of Glauber Dynamics on the Critical 2D Ising Model

**Speaker:**
Reza Gheissari

**Location:**
Warren Weaver Hall 1314

**Date:**
Thursday, March 3, 2016, 4 p.m.

**Synopsis:**

Consider the Glauber dynamics with stationary distribution given by the Ising measure on \(\mathbb Z^2\). The mixing time of the Markov chain (the time it takes to approach stationarity in \(L_1\)) is intimately related to the gap in the spectrum of its generator. We begin with some basic facts about the mixing time and spectral gap, then prove the block dynamics technique introduced by Martinelli, to recursively bound the spectral gap of a chain on a spin system. The rest of the talk will then provide a nice application of the technique to obtain a polynomial upper bound on the mixing time of the Ising model in 2D at its critical temperature (Lubetzky, Sly). If time permits we will discuss some related problems where the block dynamics could prove useful.