TY - JOUR

T1 - ENTROPY DECAY IN THE SWENDSEN–WANG DYNAMICS ON Zd

AU - Blanca, Antonio

AU - Caputo, Pietro

AU - Parisi, Daniel

AU - Sinclair, Alistair

AU - Vigoda, Eric

N1 - Funding Information:
PROOF OF THEOREM 1.3. From the discussion at the beginning of Section 8, note that there is an admissible boundary condition in the joint space for which the edge marginal is the random-cluster measure on a square region of Z2 with a wired boundary condition, and the spin marginal is the monochromatic boundary condition. The result then follows from Theorem 8.3 and Lemma 8.7. □ Funding. 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.
Publisher Copyright:
© Institute of Mathematical Statistics, 2022

PY - 2022/4

Y1 - 2022/4

N2 - 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(log n), 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.

AB - 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(log n), 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.

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U2 - 10.1214/21-AAP1702

DO - 10.1214/21-AAP1702

M3 - Article

AN - SCOPUS:85131636234

SN - 1050-5164

VL - 32

SP - 1018

EP - 1057

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 2

ER -