Entropy decay in the Swendsen-Wang dynamics on ${\mathbb Z}^d$

Abstract

We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice Z d . This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is non-local, i.e., 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 of Z d is O(log n), improving on the previous best known bound of 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. This factorization leads to several additional results, including mixing time bounds for a number of natural local and non-local Markov chains on the joint space, as well as for the standard random-cluster dynamics.

Low-temperature Ising dynamics with random initializations

“…Our main result proves that, if we initialize the dynamics on the torus in the `1 `1 `1 or ´1 ´1 ´1 configurations with probability 1 2 each (denoted by ν ˘), then the mixing time is very fast (in fact optimal). 1 Theorem 1.1.…”

“…By virtue of a comparison technique introduced in [41], the above bounds on the random-cluster dynamics translate to bounds on the Swendsen-Wang dynamics up to a multiplicative factor of N . In special settings, one can do better than this lossy comparison and obtain optimal bounds: at low temperatures this includes the complete graph [4,17,28], Z 2 [1,32] and trees [2].…”

“…More precisely, for β ă β c pdq the correlation between spins σ u , σ v goes to zero as the distance between u and v goes to infinity, and the (normalized) magnetization satisfies a central limit theorem about zero. For β ą β c pdq correlations remain uniformly bounded away from zero, and the normalized magnetization converges to a 1 2 -1 2 mixture of two point-distributions at ˘m˚p βq. We refer to Figure 1 for a schematic of this phase transition and how it relates to mixing of the Glauber dynamics.…”