Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion

TL;DR

This paper establishes optimal mixing time bounds for Glauber dynamics in various models, improving spectral independence methods and applying to graphs with bounded degree, including colorings, matchings, and spin systems.

Contribution

It introduces an improved spectral independence approach that achieves $O(n ext{log}n)$ mixing times for Glauber dynamics across multiple models on bounded degree graphs.

Findings

Proves $O(n ext{log}n)$ mixing time for Glauber dynamics on bounded degree graphs.

Establishes mixing time bounds for the hard-core model below critical fugacity.

Extends results to $q$-colorings and matchings with optimal bounds.

Abstract

We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows $O(n\log{n})$ mixing time on any $n$-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity $\lambda$, we establish $O(n\log{n})$ mixing time for the Glauber dynamics on any $n$-vertex graph of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\Delta)$ where $\lambda_c(\Delta)$ is the critical point for the uniqueness/non-uniqueness phase transition on the $\Delta$-regular tree. More generally, for any antiferromagnetic 2-spin system we prove $O(n\log{n})$ mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain $O(n\log{n})$ mixing for $q$-colorings of triangle-free graphs of maximum degree $\Delta$ when the number of colors satisfies $q > \alpha \Delta$ where $\alpha \approx 1.763$, and $O(m\log{n})$ mixing for generating random matchings of any graph with bounded degree and $m$ edges.