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November 2019 Cutoff for the Swendsen–Wang dynamics on the lattice
Danny Nam, Allan Sly
Ann. Probab. 47(6): 3705-3761 (November 2019). DOI: 10.1214/19-AOP1344

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.

Citation

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Danny Nam. Allan Sly. "Cutoff for the Swendsen–Wang dynamics on the lattice." Ann. Probab. 47 (6) 3705 - 3761, November 2019. https://doi.org/10.1214/19-AOP1344

Information

Received: 1 June 2018; Revised: 1 January 2019; Published: November 2019
First available in Project Euclid: 2 December 2019

zbMATH: 07212170
MathSciNet: MR4038041
Digital Object Identifier: 10.1214/19-AOP1344

Subjects:
Primary: 60J10 , 82C20
Secondary: 60B10 , 60K35

Keywords: Cutoff phenomenon , Markov chains , Potts model , Swendsen–Wang dynamics

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • November 2019
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