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.
Citation
Allan Sly. Yumeng Zhang. "The Glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime." Ann. Appl. Probab. 27 (5) 2646 - 2674, October 2017. https://doi.org/10.1214/16-AAP1253
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