Spatial Mixing and Systematic Scan Markov chains

TL;DR

This paper establishes a connection between strong spatial mixing (SSM) and rapid mixing of various Markov chains on lattice graphs, providing new bounds and proofs for systematic scan and Swendsen-Wang dynamics.

Contribution

It introduces a combinatorial framework linking SSM to rapid mixing, proving $O( ext{log} n)$ mixing for systematic scan and $O(1)$ relaxation time for Swendsen-Wang dynamics.

Findings

SSM implies $O( ext{log} n)$ mixing for systematic scan dynamics.

SSM implies $O(1)$ relaxation time for Swendsen-Wang dynamics.

Relaxation time of Swendsen-Wang in 2D is $O(1)$ in the subcritical regime.

Abstract

We consider spin systems on the integer lattice graph $\mathbb{Z}^d$ with nearest-neighbor interactions. We develop a combinatorial framework for establishing that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), implies rapid mixing of a large class of Markov chains. As a first application of our method we prove that SSM implies $O(\log n)$ mixing of systematic scan dynamics (under mild conditions) on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time (i.e., the inverse spectral gap). As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also use our combinatorial framework to give a simple coupling proof of the classical result that SSM entails optimal mixing time of the Glauber dynamics. Although our results in the paper focus on $d$-dimensional cubes in $\mathbb{Z}^d$, they generalize straightforwardly to arbitrary regions of $\mathbb{Z}^d$ and to graphs with subexponential growth.