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 Δ of the input graph G and hold when
k>11Δ/6 for all G. Improved results hold when k>αΔ for
graphs with girth ≥5 and Δ sufficiently large where α≈1.7632… is the root of α=exp(1/α); further improvements on
the constant α 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>αd where
α≈1.7632… is the same constant as above. Moreover we obtain
O(n3) 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