The Glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime

Abstract

The mixing time of the Glauber dynamics for spin systems on trees is closely related to the 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 [Comm. Math. Phys. 250 (2004) 301–334; Random Structures Algorithms 31 (2007) 134–172]. However, when there are hard constraints, the dynamics inside blocks 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 to essentially any spin system that has nonreconstruction 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 nonreconstruction regime.