Entropy decay in the Swendsen–Wang dynamics on Zd

Abstract

We study the mixing time of the Swendsen–Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ${\mathbb{Z}}^{\mathit{d}}$. 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 ${\mathbb{Z}}^{\mathit{d}}$ is $\mathit{O}(log\mathit{n})$, and we prove this is tight by establishing a matching lower bound on the mixing time. The previous best known bound was $\mathit{O}(\mathit{n})$. SSM is a standard condition corresponding to exponential decay of correlations with distance between spins on the lattice and is known to hold in $\mathit{d}=2$ 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.