The Mixing Time of Glauber Dynamics for Colouring Regular Trees

TL;DR

This paper analyzes the mixing time of Glauber dynamics for proper q-colorings on regular trees, providing bounds that depend on the number of colors and the tree's branching factor.

Contribution

It establishes both upper and lower bounds on the mixing time for Glauber dynamics in coloring regular trees, advancing understanding of sampling efficiency.

Findings

Upper bound on mixing time: n^{O(b/\log b)}

Lower bound on mixing time: n^{\Omega(b/q \log b)}

Bounds depend on q and b, with constants independent of n, q, and b

Abstract

We consider Metropolis Glauber dynamics for sampling proper $q$-colourings of the $n$-vertex complete $b$-ary tree when $3\leq q\leq b/2\ln(b)$. 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 $b$.