Spatial Mixing and Non-local Markov chains

TL;DR

This paper demonstrates that strong spatial mixing (SSM) in spin systems leads to rapid convergence of various non-local Markov chains, including block and Swendsen-Wang dynamics, with bounds independent of system size.

Contribution

It establishes new theoretical links between spatial mixing conditions and mixing times of non-local Markov chains, extending previous results and providing practical bounds.

Findings

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

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

Systematic scan dynamics have $O( ext{log} n ( ext{log} ext{log} n)^2)$ mixing time under SSM.

Abstract

We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. 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. 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 prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)^2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.