The Annals of Probability

Cutoff for the Swendsen–Wang dynamics on the lattice

Danny Nam and Allan Sly

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Abstract

We study the Swendsen–Wang dynamics for the $q$-state Potts model on the lattice. Introduced as an alternative algorithm of the classical single-site Glauber dynamics, the Swendsen–Wang dynamics is a nonlocal Markov chain that recolors many vertices at once based on the random-cluster representation of the Potts model. In this work, we establish cutoff phenomenon for the Swendsen–Wang dynamics on the lattice at sufficiently high temperatures, proving that it exhibits a sharp transition from “unmixed” to “well mixed.” In particular, we show that at high enough temperatures the Swendsen–Wang dynamics on the torus $(\mathbb{Z}/n\mathbb{Z})^{d}$ has cutoff at time $\frac{d}{2}(-\log (1-\gamma ))^{-1}\log n$, where $\gamma (\beta )$ is the spectral gap of the infinite-volume dynamics.

Article information

Source
Ann. Probab., Volume 47, Number 6 (2019), 3705-3761.

Dates
Received: June 2018
Revised: January 2019
First available in Project Euclid: 2 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1575277340

Digital Object Identifier
doi:10.1214/19-AOP1344

Mathematical Reviews number (MathSciNet)
MR4038041

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B10: Convergence of probability measures

Keywords
Markov chains Swendsen–Wang dynamics Potts model cutoff phenomenon

Citation

Nam, Danny; Sly, Allan. Cutoff for the Swendsen–Wang dynamics on the lattice. Ann. Probab. 47 (2019), no. 6, 3705--3761. doi:10.1214/19-AOP1344. https://projecteuclid.org/euclid.aop/1575277340


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