The Annals of Probability
- Ann. Probab.
- Volume 41, Number 1 (2013), 294-328.
Exact thresholds for Ising–Gibbs samplers on general graphs
We establish tight results for rapid mixing of Gibbs samplers for the Ferromagnetic Ising model on general graphs. We show that if
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
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.
Ann. Probab., Volume 41, Number 1 (2013), 294-328.
First available in Project Euclid: 23 January 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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