Rapid Mixing of Glauber Dynamics for Monotone Systems via Entropic Independence

TL;DR

This paper establishes rapid mixing times for Glauber dynamics in certain monotone systems using entropic independence, improving bounds for models like the ferromagnetic Ising and bipartite hardcore models.

Contribution

It introduces a new mixing time comparison for Glauber dynamics under entropic independence and applies it to improve bounds for specific models.

Findings

O(tilde)(n) mixing time for ferromagnetic Ising model with biased fields

O(tilde)(n^2) mixing time for bipartite hardcore model under one-sided uniqueness

New comparison result between Glauber and field dynamics

Abstract

We study the mixing time of Glauber dynamics on monotone systems. For monotone systems satisfying the entropic independence condition, we prove a new mixing time comparison result for Glauber dynamics. For concrete applications, we obtain $\tilde{O}(n)$ mixing time for the random cluster model induced by the ferromagnetic Ising model with consistently biased external fields, and $\tilde{O}(n^2)$ mixing time for the bipartite hardcore model under the one-sided uniqueness condition, where $n$ is the number of variables in corresponding models, improving the best known results in [Chen and Zhang, SODA'23] and [Chen, Liu, and Yin, FOCS'23], respectively. Our proof combines ideas from the stochastic dominance argument in the classical censoring inequality and the recently developed high-dimensional expanders. The key step in the proof is a novel comparison result between the Glauber dynamics and the field dynamics for monotone systems.