Improved Mixing of Critical Hardcore Model

TL;DR

This paper demonstrates that the Glauber dynamics for the critical hardcore model mixes faster than previously known, with a polynomial bound that improves upon earlier results, by establishing optimal spectral independence bounds.

Contribution

It provides the first polynomial mixing time bound for the critical hardcore model at the phase transition, improving previous bounds by leveraging spectral independence techniques.

Findings

Glauber dynamics mixes in O(n^{4+O(1/elta)}) time at criticality.

Improves previous upper bound of O(n^{12.88+O(1/elta)}).

Establishes optimal spectral independence bounds at the phase transition.

Abstract

The hardcore model is one of the most classic and widely studied examples of undirected graphical models. Given a graph $G$, the hardcore model describes a Gibbs distribution of $\lambda$-weighted independent sets of $G$. In the last two decades, a beautiful computational phase transition has been established at a precise threshold $\lambda_c(\Delta)$ where $\Delta$ denotes the maximum degree, where the task of sampling independent sets transitions from polynomial-time solvable to computationally intractable. We study the critical hardcore model where $\lambda = \lambda_c(\Delta)$ and show that the Glauber dynamics, a simple yet popular Markov chain algorithm, mixes in $\tilde{O}(n^{4+O(1/\Delta)})$ time on any $n$-vertex graph of maximum degree $\Delta\geq3$, significantly improving the previous upper bound $\tilde{O}(n^{12.88+O(1/\Delta)})$ by the recent work arXiv:2411.03413. Our improvement comes from an optimal bound on the $\ell_\infty$-spectral independence for the hardcore model at all subcritical fugacity $\lambda < \lambda_c(\Delta)$.