The Annals of Applied Probability

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

Allan Sly and Yumeng Zhang

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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.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 2646-2674.

Dates
Received: January 2015
Revised: July 2016
First available in Project Euclid: 3 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1509696031

Digital Object Identifier
doi:10.1214/16-AAP1253

Mathematical Reviews number (MathSciNet)
MR3719943

Zentralblatt MATH identifier
1379.60079

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Mixing time Glauber dynamics graph colorings reconstruction problem

Citation

Sly, Allan; Zhang, Yumeng. The Glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime. Ann. Appl. Probab. 27 (2017), no. 5, 2646--2674. doi:10.1214/16-AAP1253. https://projecteuclid.org/euclid.aoap/1509696031


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