Sly, AllanZhang, Yumeng2021-09-292021-09-291050-5164http://hdl.handle.net/1885/248816The 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(nlog n) in the entire nonreconstruction regime.application/pdfen-AU© Institute of Mathematical Statistics, 2017Mixing timeGlauber dynamicsgraph coloringsreconstruction problem201710.1214/16-AAP12532020-11-23