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

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.