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 λ, we establish O(n log n) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree ∆ when λ < λ c (∆) where λ c (∆) is the critical point for the uniqueness/non-uniqueness phase transition on the ∆-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 ∆ when the number of colors satisfies q > α∆ where α ≈ 1.763, and O(m log n) 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).

Optimal Mixing Time for the Ising Model in the Uniqueness Regime

“…We now give an outline of the proof of Lemma 2.6. First, we show that for all sufficiently large , the distribution satisfies the uniform block factorization of entropy in [CLV21a]. Next, we show that when → ∞, the uniform block factorization of entropy for implies the magnetized block factorization of entropy for the original distribution .…”

“…By (10), we can view as a distribution over ( ) = → ( ) for all 0 ≤ < . The following lemma is proved in [CLV21a] (see the proof of Lemma 2.6 in the full version of [CLV21a]).…”

“…Recently, based on a powerful "local-to-global" theorem of Alev and Lau [ALO20, AL20] for high-dimensional expander walks, an important concept called spectral independence was introduced by Anari, Liu and Oveis Gharan in their seminal work [ALO20], which leads to a series of important progress in studies of mixing times [CLV20, FGYZ21, CGŠV21, Liu21, BCC + 21, JPV21, CFYZ21, CLV21b, ALG21, AJK + 21]. For the Ising models in the uniqueness regime with ∈ [ Δ−2+ Δ− , Δ− Δ−2+ ], a polynomially-bounded mixing time (1/ ) was first proved by Chen, Liu and Vigoda [CLV20] using the spectral independence; and this mixing time bound was improved to Δ (1/ ) log in their subsequent work [CLV21a] by a uniform block factorization of entropy. On general graphs with unbounded maximum degrees, both these mixing bounds become (1/ ) in the worst case.…”

“…Lemma 8.4 ([CLV21a]). Let be the Gibbs distribution of Ising model on graph = ( , ) with maximum degree Δ ≥ 3, edge activity and local fields = ( ) ∈ .…”

Simple Parallel Algorithms for Single-Site Dynamics

“…Assuming uniqueness condition, an optimal O(n log n)-step discrete mixing time was proved for the Glauber dynamics on graphs with bounded maximum degree [CLV21,MS13], and very recently on all graphs [CFYZ21]. Furthermore, Condition 1 holds easily for the Ising model for any…”