  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.

Probability;Combinatorics