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 Erd o s-R 'enyi random graph G(n,d n). While the average degree in G(n,d n) is d(1-o(1)), it contains many nodes of degree of order @math . The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring or the Ising model, the mixing time of Gibbs sampling is at least @math . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring. In this work consider sampling q-colorings and show that for every $d 1 where the number of colors does not depend on n. They extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. The results also generalize to the hard-core model at low fugacity and to general models of soft constraints at high temperatures.
We note that for all @math and all @math the mixing time of Gibbs sampling on @math is with high probability at least @math , see @cite_9 @cite_20 for details. It is an important challenge to find the critical @math for rapid mixing. In particular, the question is if the threshold can be formulated in terms of the coloring model on a branching process tree with @math degree distribution. One would expect rapid mixing for in the uniqueness phase'', but perhaps even beyond it, see @cite_10 @cite_20 @cite_5 .
@cite_5 @cite_9 @cite_10 @cite_20
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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 β < 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 Cnlogn. 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 dtanh β < 1, with high probability over the Erdős–Renyi random graph G(n,d n), it holds that the mixing time of Gibbs samplers is n1+Θ(1 loglogn). Both results are tight, as it is known that the mixing time for random regular and Erdős–Renyi random graphs is, with high probability, exponential in n when (d−1) tanh β> 1, and dtanh β>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. In this work we show that for every @math , such that for all @math where the parameters of the model do not depend on @math . 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 @math . Our proof exploits in novel ways the local treelike structure of Erd o s-R 'enyi 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 @math of the graph has a neighborhood @math of radius @math in which the induced sub-graph is a tree union at most @math edges and where for each simple path in @math the sum of the vertex degrees along the path is @math . 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 @math it applies for all external fields and @math , where @math is the critical point for decay of correlation for the Ising model on @math . We consider local Markov chain Monte–Carlo algorithms for sampling from the weighted distribution of independent sets with activity λ, where the weight of an independent set I is λ|I|. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree d and λ λ c it is NP-hard to approximate the above weighted sum over independent sets to within a factor polynomial in the size of the graph. Consider k-colorings of the complete tree of depth l and branching factor Δ. If we fix the coloring of the leaves, for what range of k is the root uniformly distributed over all k colors (in the limit l→∞)? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which “freeze” the entire tree when k≤Δ+1. For k≥Δ+2, Jonasson proved the root is “unbiased” for any fixed coloring of the leaves, and thus the Gibbs measure is unique. What happens for a typical coloring of the leaves? When the leaves have a nonvanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Nonreconstruction is equivalent to extremality of the free-boundary Gibbs measure. When k 1 and k=CΔ lnΔ, nonreconstruction holds; i.e., the Gibbs measure is extr...