Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

TL;DR

This paper analyzes the phase transition in the mixing time of Glauber dynamics for coloring regular trees, identifying a sharp threshold at a critical number of colors related to the reconstruction threshold.

Contribution

It establishes nearly sharp bounds on the mixing time for random k-colorings of regular trees, revealing a phase transition at a critical ratio of colors to branching factor.

Findings

For C ≥ 1, mixing time is O(n^{1+o_b(1)} log n).

For C < 1, mixing time slows down to O(n^{1/C+o_b(1)} log n).

Critical point C=1 aligns with the reconstruction threshold.

Abstract

We prove that the mixing time of the Glauber dynamics for random k-colorings of the complete tree with branching factor b undergoes a phase transition at $k=b(1+o_b(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with n vertices for $k=Cb/\ln{b}$ colors with constant C. For $C\geq1$ we prove the mixing time is $O(n^{1+o_b(1)}\ln{n})$. On the other side, for $C<1$ the mixing time experiences a slowing down; in particular, we prove it is $O(n^{1/C+o_b(1)}\ln{n})$ and $\Omega(n^{1/C-o_b(1)})$. The critical point C=1 is interesting since it coincides (at least up to first order) with the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.