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- W2743402684 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." @default.
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- W2743402684 date "2017-08-03" @default.
- W2743402684 modified "2023-09-27" @default.
- W2743402684 title "Spatial Mixing and Non-local Markov chains" @default.
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