# Mathematics Colloquia and Seminars

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.