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 $(d-1) \tanh \beta < 1$, then there exists a constant $C$ such that the discrete time mixing time of Gibbs Samplers for the Ferromagnetic Ising model on {\em any} graph of $n$ vertices and maximal degree $d$, where all interactions are bounded by $\beta$, and arbitrary external fields is bounded by $C n \log n$. We further show the when $d \tanh \beta < 1$, with high probability over the Erd\H{o}s-R\'enyi random graph on $n$ vertices with average degree $d$, it holds that the mixing time of Gibbs Samplers is $n^{1+\Theta(\frac{1}{\log \log n})}$. Both result are tight as it is known that the mixing time for random regular and Erd\H{o}s-R\'enyi random graphs is, with high probability, exponential in $n$ when if $(d-1) \tanh \beta > 1$ and $d \tanh \beta > 1$ respectively.