## The Annals of Applied Probability

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

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

#### 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