Rapid mixing of Gibbs sampling on graphs that are sparse on average

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erdös-Rényi random graph <i>G</i>(<i>n, d/n</i>), where each edge is chosen independently with probability <i>d/n</i> and <i>d</i> is fixed. While the average degree in <i>G</i>(<i>n, d/n</i>) is <i>d</i>(1 - <i>o</i>(1)), it contains many nodes of degree of order log <i>n</i>/ log log <i>n.</i> The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on <i>G(n, p)</i> such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least <i>n</i>^{1+Ω(1/log log <i>n</i>)}. Recall that the Ising model with inverse temperature β defined on a graph <i>G</i> = (<i>V, E</i>) is the distribution over {±}^V given by <i>P</i>(σ) = 1/Z exp (β Σ_{<i>(v,u)∈E</i>} σ<i>(v)σ(u)).</i> High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of β or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work we show that for every <i>d</i> < ∞ and the Ising model defined on <i>G(n, d/n)</i>, there exists a β<i>_d</i> > 0, such that for all β < β<i>_d</i> with probability going to 1 as <i>n</i> → ∞, the mixing time of the dynamics on <i>G(n, d/n)</i> is polynomial in <i>n.</i> Our results are the first polynomial time mixing results proven for a natural model on <i>G(n, d/n)</i> for <i>d</i> > 1 where the parameters of the model do not depend on <i>n</i>. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than <i>n</i> polylog<i>(n)</i>. Our proof exploits in novel ways the local treelike structure of Erdős-Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex <i>v</i> of the graph has a neighborhood <i>N (v)</i> of radius <i>O</i> (log <i>n</i>) in which the induced subgraph is a tree union at most <i>O</i>(log <i>n</i>) edges and where for each simple path in <i>N(v)</i> the sum of the vertex degrees along the path is <i>O</i>(log <i>n</i>). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for <i>G(n, d/n)</i> it applies for all external fields and β < β<i>d</i>, where <i>d</i> tanh(β<i>d</i>) = 1 is the critical point for decay of correlation for the Ising model on <i>G(n, d/n).</i>