Rapid mixing of Glauber dynamics for colorings below Vigoda's $11/6$ threshold

Abstract

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of $k$-colorings of a graph $G$ on $n$ vertices with maximum degree $\Delta$ is rapidly mixing for $k \geq \Delta +2$. In FOCS 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper $k$-colorings for $k > \frac{11}{6}\Delta$, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the $\frac{11}{6}\Delta$ barrier for general graphs by showing rapid mixing for $k > (\frac{11}{6} - \eta)\Delta$ for some positive constant $\eta$. 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.