Sly, Allan et Zhang, Yumeng, The glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime, Annals of applied probability , 27(5), 2017, pp. 2646-2674
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.