April 2022 Entropy decay in the Swendsen–Wang dynamics on Zd
Antonio Blanca, Pietro Caputo, Daniel Parisi, Alistair Sinclair, Eric Vigoda
Author Affiliations +
Ann. Appl. Probab. 32(2): 1018-1057 (April 2022). DOI: 10.1214/21-AAP1702


We study the mixing time of the Swendsen–Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice Zd. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is nonlocal, that is, it changes the entire configuration in one step. We prove that, whenever strong spatial mixing (SSM) holds, the mixing time on any n-vertex cube in Zd is O(logn), and we prove this is tight by establishing a matching lower bound on the mixing time. The previous best known bound was O(n). SSM is a standard condition corresponding to exponential decay of correlations with distance between spins on the lattice and is known to hold in d=2 dimensions throughout the high-temperature (single phase) region. Our result follows from a modified log-Sobolev inequality, which expresses the fact that the dynamics contracts relative entropy at a constant rate at each step. The proof of this fact utilizes a new factorization of the entropy in the joint probability space over spins and edges that underlies the Swendsen–Wang dynamics, which extends to general bipartite graphs of bounded degree. This factorization leads to several additional results, including mixing time bounds for a number of natural local and nonlocal Markov chains on the joint space, as well as for the standard random-cluster dynamics.

Funding Statement

The first author was supported in part by NSF Grant CCF-1850443.
The fourth author was supported in part by NSF Grant CCF-1815328.
The fifth author was supported in part by NSF Grant CCF-2007022.


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Antonio Blanca. Pietro Caputo. Daniel Parisi. Alistair Sinclair. Eric Vigoda. "Entropy decay in the Swendsen–Wang dynamics on Zd." Ann. Appl. Probab. 32 (2) 1018 - 1057, April 2022. https://doi.org/10.1214/21-AAP1702


Received: 1 July 2020; Revised: 1 March 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

MathSciNet: MR4414700
zbMATH: 1487.60131
Digital Object Identifier: 10.1214/21-AAP1702

Primary: 60J10 , 68Q87 , 82B20

Keywords: Log-Sobolev , mixing time , Potts model , Swendsen–Wang

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 2 • April 2022
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