Rapid Mixing of Glauber Dynamics via Spectral Independence for All Degrees

Abstract

SIAM Journal on Computing, Ahead of Print.
Abstract. We prove an optimal [math] lower bound on a spectral gap of the Glauber dynamics for antiferromagnetic two-spin systems with [math] vertices in the tree uniqueness regime. This spectral gap holds for any, including unbounded, maximum degree [math]. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack [math]: [math] mixing time for the hardcore model with fugacity [math] and [math] mixing time for the Ising model with edge activity [math], where the maximum degree [math] may depend on the number of vertices [math] and [math] depends only on [math]. Our proof is built on the recently developed connections between the Glauber dynamics for spin systems and high-dimensional expander walks. In particular, we prove a stronger notion of spectral independence, called complete spectral independence, and use a novel Markov chain, called field dynamics, to connect this stronger spectral independence to the rapid mixing of Glauber dynamics for all degrees.