Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion

Abstract

We prove an optimal mixing time bound for 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(nlogn) 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 ?, we establish O(nlogn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree ?when ?? ?where ? ? 1.763, and O(mlogn) mixing for generating random matchings of any graph with bounded degree and m edges. Our approach is based on two steps. First, we show that the approximate tensorization of entropy (i.e., factorizing entropy into single vertices), which is a key step for establishing the modified log-Sobolev inequality in many previous works, can be deduced from entropy factorization into blocks of fixed linear size. Second, we adapt the local-to-global scheme of Alev and Lau (2020) to establish such block factorization of entropy in a more general setting of pure weighted simplicial complexes satisfying local spectral expansion; this also substantially generalizes the result of Cryan et al. (2019).