Kawasaki dynamics beyond the uniqueness threshold

TL;DR

This paper demonstrates that Kawasaki dynamics for the ferromagnetic Ising model on large random regular graphs mixes rapidly beyond the known tree uniqueness threshold, extending understanding of mixing times in statistical physics models.

Contribution

It establishes a spectral condition for fast mixing of conservative Ising dynamics beyond the tree uniqueness threshold, applicable to large regular graphs.

Findings

Kawasaki dynamics mixes fast beyond the tree uniqueness threshold for large d.

Spectral condition for log-Sobolev inequalities in conservative Ising models.

Extension of results to perturbations with log-concave generating polynomials.

Abstract

Glauber dynamics of the Ising model on a random regular graph is known to mix fast below the tree uniqueness threshold and exponentially slowly above it. We show that Kawasaki dynamics of the canonical ferromagnetic Ising model on a random $d$-regular graph mixes fast beyond the tree uniqueness threshold when $d$ is large enough (and conjecture that it mixes fast up to the tree reconstruction threshold for all $d\geq 3$). This result follows from a more general spectral condition for (modified) log-Sobolev inequalities for conservative dynamics of Ising models. The proof of this condition in fact extends to perturbations of distributions with log-concave generating polynomial.