Exact thresholds for Ising-Gibbs samplers on general graphs

TL;DR

This paper establishes precise conditions under which Gibbs samplers for the ferromagnetic Ising model mix rapidly on general graphs, providing the first tight thresholds for such dynamics and their relation to spatial thresholds on trees.

Contribution

The paper provides the first tight sufficient conditions for rapid mixing of Gibbs samplers on general graphs and links these thresholds to spatial thresholds on trees.

Findings

Rapid mixing when (d-1) tanh β < 1 on graphs of degree d.

Spectral gap remains bounded away from zero under these conditions.

Mixing time on Erdős-Rényi graphs is nearly polynomial when d tanh β < 1.

Abstract

We establish tight results for rapid mixing of Gibbs samplers for the Ferromagnetic Ising model on general graphs. We show that if \[(d-1)\tanh\beta<1,\] then there exists a constant C such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of n vertices and maximal degree d, where all interactions are bounded by $\beta$, and arbitrary external fields are bounded by $Cn\log n$. Moreover, the spectral gap is uniformly bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d. We further show that when $d\tanh\beta<1$, with high probability over the Erdos-Renyi random graph $G(n,d/n)$, it holds that the mixing time of Gibbs samplers is \[n^{1+\Theta({1}/{\log\log n})}.\] Both results are tight, as it is known that the mixing time for random regular and Erdos-Renyi random graphs is, with high probability, exponential in n when $(d-1)\tanh\beta>1$, and $d\tanh\beta>1$, respectively. To our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs. Moreover, our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees.