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Rapid mixing of Glauber dynamics for colorings below Vigoda's threshold
A well-known conjecture in computer science and statistical physics is that
Glauber dynamics on the set of -colorings of a graph on vertices
with maximum degree is rapidly mixing for . In FOCS
1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters
on the set of proper -colorings for , implying rapid
mixing for Glauber dynamics. In this paper, we obtain the first improvement
beyond the barrier for general graphs by showing rapid
mixing for for some positive constant .
The key to our proof is combining path coupling with a new kind of metric that
incorporates a count of the extremal configurations of the chain. Additionally,
our results extend to list coloring, a widely studied generalization of
coloring. Combined, these results answer two open questions from Frieze and
Vigoda's 2007 survey paper on Glauber dynamics for colorings.Comment: 21 page