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, Liu, and Oveis Gharan [Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 1319–1330] and shows mixing time on any -vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hardcore model on independent sets weighted by a fugacity , we establish mixing time for the Glauber dynamics on any -vertex graph of constant maximum degree when , where is the critical point for the uniqueness/nonuniqueness phase transition on the -regular tree. More generally, for any antiferromagnetic 2-spin system we prove the 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 mixing for -colorings of triangle-free graphs of maximum degree when the number of colors satisfies , where , and mixing for generating random matchings of any graph with bounded degree and 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 [Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), 2020, pp. 1198–1211] 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, Guo, and Mousa, [Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2019, pp. 1358–1370].