Spatial Mixing and Systematic Scan Markov chains

We consider spin systems on the integer lattice graph Z 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 ddimensional cube of the integer lattice graph Z. 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 Z is O(1) throughout the subcritical regime of the q-state Potts model, for all q ≥ 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 Z, they generalize straightforwardly to arbitrary regions of Z and to graphs with subexponential growth. School of Computer Science, Georgia Tech, Atlanta, GA 30332. Email: [email protected]. Research supported in part by NSF grants 1420934, 1555579 and 1617306. Department of Mathematics, University of Roma Tre, Largo San Murialdo 1, 00146 Roma, Italy. Email: [email protected] Computer Science Division, U.C. Berkeley, Berkeley, CA 94720. Email: [email protected]. Research supported in part by NSF grant 1420934. School of Computer Science, Georgia Tech, Atlanta, GA 30332. Email: [email protected]. Research supported in part by NSF grants 1555579 and 1617306. ∗∗ Part of this work was done at the Simons Institute for the Theory of Computing.