# Improved bounds for sampling colorings of sparse random graphs

We study the mixing properties of the single-site Markov chain known
as the Glauber dynamics for sampling k-colorings of a sparse random
graph G(n,d/n) for constant d. The best known rapid mixing results
for general graphs are in terms of the maximum degree Delta of the
input graph G and hold when k>11Delta/6 for all G. Improved results
hold when k>alphaDelta for graphs with girth =>5 and Delta
sufficiently large where alphaapprox 1.7632… is the root of
alpha=exp(1/alpha); further improvements on the constant alpha
hold with stronger girth and maximum degree assumptions.

For sparse random graphs the maximum degree is a function of n and the
goal is to obtain results in terms of the expected degree d. The
following rapid mixing results for G(n,d/n) hold with high probability
over the choice
of the random graph for sufficiently large constant d. Mossel and
Sly (2009) proved rapid mixing for constant k, and Efthymiou (2014)
improved this to k linear in~d. The condition was improved to k>3d by
Yin and Zhang (2016) using non-MCMC methods.

Here we prove rapid mixing when k>alpha d where \$alphaapprox
1.7632… is the same constant as above. Moreover we obtain O(n^{3})
mixing time of the Glauber dynamics, while in previous rapid mixing results
the exponent was an increasing function in d. As in previous results
for random graphs our proof analyzes an appropriately defined block
dynamics to “hide’’ high-degree vertices. One new aspect in our
improved approach is utilizing so-called local uniformity properties
for the analysis of block dynamics. To analyze the “burn-in’’ phase
we prove a concentration inequality for the number of disagreements
propagating in large blocks.

This is a joint work with Tom Hayes, Daniel Stefankovic and Eric Vigoda.