Glauber Dynamics of colorings on trees

TL;DR

This paper extends the analysis of Glauber dynamics for spin systems on trees to include systems with hard constraints, establishing polynomial mixing times in the non-reconstruction regime.

Contribution

It introduces a new variant of block dynamics applicable to hard-constrained spin systems, broadening previous results that were limited to soft constraints.

Findings

Mixing time for colorings on regular trees is O(n log n) in the non-reconstruction regime.

The new dynamics apply to a wide class of systems with hard constraints.

Results connect non-reconstruction with rapid mixing in these systems.

Abstract

The mixing time of the Glauber dynamics for spin systems on trees is closely related to reconstruction problem. Martinelli, Sinclair and Weitz established this correspondence for a class of spin systems with soft constraints bounding the log-Sobolev constant by a comparison with the block dynamics. However, when there are hard constraints, the block dynamics may be reducible. We introduce a variant of the block dynamics extending these results to a wide class of spin systems with hard constraints. This applies for essentially any spin system that has non-reconstruction provided that on average the root is not locally frozen in a large neighborhood. In particular we prove that the mixing time of the Glauber dynamics for colorings on the regular tree is $O(n\log n)$ in the entire known non-reconstruction regime.