A simple proof of rapid mixing on random regular graphs beyond uniqueness

2026-06-25

AI Summary

This work investigates the rapid mixing of Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. By ingeniously adapting the Bochner–Bakry–Émery method—originally developed for continuous-space Gibbs point processes—to the discrete setting, the authors directly establish a Poincaré inequality. Their approach leverages an expansion of the Dirichlet form and eliminates squared-sum terms, thereby circumventing intricate recursive arguments or coupling constructions. This yields a concise, self-contained proof of rapid mixing that requires no additional assumptions, significantly streamlining the existing theoretical framework. Notably, it provides the first explicit lower bound on the spectral gap in the non-uniqueness regime.

Abstract

A recent breakthrough of Chen, Chen, Chen, Yin, and Zhang shows rapid mixing for Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. Their approach builds upon the literature of various local-to-global techniques and applies to a more general setting of discrete distributions supported on downward-closed set families. We give a short and self-contained proof via a Bochner--Bakry--Émery approach and directly show 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 of Kondratiev, Kuna, and Ohlerich used to study spatial birth-and-death dynamics for Gibbs point processes in the continuum, which we adapt to the discrete setting.

Problem

• rapid mixing
• hard-core model
• random regular graphs
• Glauber dynamics
• uniqueness threshold

Innovation

• rapid mixing
• Glauber dynamics
• Poincaré inequality
• Bochner–Bakry–Émery method
• hard-core model