Abstract
A recent breakthrough by Chen et al. shows rapid mixing for Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. Their approach builds on various local-to-global techniques and applies to a more general setting of discrete distributions supported on downward-closed set families.
We provide a short and self-contained proof via a Bochner-Bakry-Émery approach, directly demonstrating a Poincaré inequality by expanding the Dirichlet form in terms of the $L^2$-norm of the generator applied to a test function and eliminating a sum of squares term. Our proof is a streamlined version of an argument by Kondratiev, Kuna, and Ohlerich used to study spatial birth-and-death dynamics for Gibbs point processes in the continuum, adapted to the discrete setting.
Blogger's Review: This paper presents a novel proof of rapid mixing for Glauber dynamics on random regular graphs using the Bochner-Bakry-Émery method, highlighting the broad applicability of local-to-global techniques in discrete distributions. Its implications are significant for both theoretical exploration and practical applications.