Gibbs Rapidly Samples Colorings of G(n,d/n)

TL;DR

This paper proves that Gibbs sampling for q-colorings on Erdős-Rényi graphs G(n,d/n) mixes in polynomial time for sufficiently large q, extending to broader sparse graph models and other statistical physics models.

Contribution

It establishes the first polynomial mixing time results for coloring models on G(n,d/n) with a fixed number of colors, independent of n.

Findings

Gibbs sampling mixes rapidly for large enough q on G(n,d/n)

Results extend to sparse graphs with average degree constraints

Applicable to other models like the hard-core and soft constraint models

Abstract

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\H{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 $\log n / \log \log n$. 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 $n^{1 + \Omega(1 / \log \log n)}$. 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 $d < \infty$ there exists $q(d) < \infty$ such that for all $q \geq q(d)$ the mixing time of Gibbs sampling on G(n,d/n) is polynomial in $n$ with high probability. Our results are the first polynomial time mixing results proven for the coloring model on G(n,d/n) for 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.