Open Access
February 2020 Random-cluster dynamics in Z2: Rapid mixing with general boundary conditions
Antonio Blanca, Reza Gheissari, Eric Vigoda
Ann. Appl. Probab. 30(1): 418-459 (February 2020). DOI: 10.1214/19-AAP1505

Abstract

The random-cluster model with parameters (p,q) is a random graph model that generalizes bond percolation (q=1) and the Ising and Potts models (q2). We study its Glauber dynamics on n×n boxes Λn of the integer lattice graph Z2, where the model exhibits a sharp phase transition at p=pc(q). Unlike traditional spin systems like the Ising and Potts models, the random-cluster model has non-local interactions. Long-range interactions can be imposed as external connections in the boundary of Λn, known as boundary conditions. For select boundary conditions that do not carry long-range information (namely, wired and free), Blanca and Sinclair proved that when q>1 and ppc(q), the Glauber dynamics on Λn mixes in optimal O(n2logn) time. In this paper, we prove that this mixing time is polynomial in n for every boundary condition that is realizable as a configuration on Z2Λn. We then use this to prove near-optimal O~(n2) mixing time for “typical” boundary conditions. As a complementary result, we construct classes of nonrealizable (nonplanar) boundary conditions inducing slow (stretched-exponential) mixing at ppc.

Citation

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Antonio Blanca. Reza Gheissari. Eric Vigoda. "Random-cluster dynamics in : Rapid mixing with general boundary conditions." Ann. Appl. Probab. 30 (1) 418 - 459, February 2020. https://doi.org/10.1214/19-AAP1505

Information

Received: 1 July 2018; Revised: 1 May 2019; Published: February 2020
First available in Project Euclid: 25 February 2020

zbMATH: 07200532
MathSciNet: MR4068315
Digital Object Identifier: 10.1214/19-AAP1505

Subjects:
Primary: 60K35
Secondary: 82B20 , 82C20

Keywords: Glauber dynamics , mixing time , Random-cluster model , spatial mixing

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.30 • No. 1 • February 2020
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