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 (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\logn. Moreover, the spectral gap is uniformly
bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d …
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\logn. Moreover, the spectral gap is uniformly
bounded away from 0 for all such graphs, as well as for infinite graphs of maximal degree d …
Abstract
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 such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of vertices and maximal degree , where all interactions are bounded by , and arbitrary external fields are bounded by . Moreover, the spectral gap is uniformly bounded away from for all such graphs, as well as for infinite graphs of maximal degree .
We further show that when , with high probability over the Erdős–Rényi random graph , 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 when , and , 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.
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