Spatial Mixing and Systematic Scan Markov chains
(1612.01576)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.
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