
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
Recommended Citation
Mossel, E., & Sly, A. (2013). Exact Thresholds for Ising–Gibbs Samplers on General Graphs. The Annals of Probability, 41 (1), 294-328. http://dx.doi.org/10.1214/11-AOP737
Date Posted: 27 November 2017
This document has been peer reviewed.