The Mixing Time of Glauber Dynamics for Colouring Regular Trees

Abstract

We consider Metropolis Glauber dynamics for sampling proper $q$-colourings of the $n$-vertex complete $b$-ary tree when $3\le q\le b/2\ln (b)$. We give both upper and lower bounds on the mixing time. For fixed $q$ and $b$, our upper bound is $n^{O(b/\log b)}$ and our lower bound is $n^{\Omega (b/q\log (b))}$, where the constants implicit in the $O()$ and $\Omega ()$ notation do not depend upon $n$, $q$ or $b$.