Statistics Papers

Document Type

Journal Article

Date of this Version

2013

Publication Source

The Annals of Probability

Volume

41

Issue

1

Start Page

294

Last Page

328

DOI

10.1214/11-AOP737

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) tanhβ < 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 β, 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 β < 1, with high probability over the Erdős–Rényi random graph G(n,d/n), it holds that the mixing time of Gibbs samplers is

n1 + Θ(1/loglogn).

Both results are tight, as it is known that the mixing time for random regular and Erdős–Rényi random graphs is, with high probability, exponential in n when (d − 1) tanh β > 1, and d tanh β > 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.

Keywords

Ising model, Glauber dynamics, phase transition

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Date Posted: 27 November 2017

This document has been peer reviewed.