Polynomial Mixing of the critical Glauber Dynamics for the Ising Model

TL;DR

This paper proves that the mixing time of the Glauber Dynamics for the Ising Model at the critical threshold on graphs with bounded degree is at most polynomial in the number of vertices, using new log-Sobolev bounds and a self-avoiding walk tree construction.

Contribution

It introduces a simple proof that the mixing time at the critical point is polynomial, combining new log-Sobolev bounds with a self-avoiding walk tree approach.

Findings

Mixing time at critical threshold is polynomial in graph size.

New log-Sobolev bounds are established.

The proof is simpler than previous methods.

Abstract

In this note, we prove that on any graph of maximal degree $d$ the mixing time of the Glauber Dynamics for the Ising Model at $\beta_c=\tanh^{-1}(\frac1{d-1})$, the uniqueness threshold on the infinite $d$-regular tree, is at most polynomial in $n$. The proof follows by a simple combination of new log-Sobolev bounds of Bauerschmidt and Dagallier, together with the tree of self avoiding walks construction of Weitz. While preparing this note we became aware that Chen, Chen, Yin and Zhang recently posted another proof of this result. We believe the simplicity of our argument is of independent interest.