The Annals of Probability

Exact thresholds for Ising–Gibbs samplers on general graphs

Elchanan Mossel and Allan Sly

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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 Erdős–Rényi 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 Erdős–Rényi 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.

Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 294-328.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1358951988

Digital Object Identifier
doi:10.1214/11-AOP737

Mathematical Reviews number (MathSciNet)
MR3059200

Zentralblatt MATH identifier
1270.60113

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Keywords
Ising model Glauber dynamics phase transition

Citation

Mossel, Elchanan; Sly, Allan. Exact thresholds for Ising–Gibbs samplers on general graphs. Ann. Probab. 41 (2013), no. 1, 294--328. doi:10.1214/11-AOP737. https://projecteuclid.org/euclid.aop/1358951988


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