Rapid Mixing of Glauber Dynamics for Monotone Systems via Entropic Independence

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.