Spatial mixing and nonlocal Markov chains

We consider spin systems with nearest-neighbor interactions on an n-vertex d-dimensional cube of the integer lattice graph (Formula presented.). We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium of nonlocal Markov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. As a second application of our technology, it is established that SSM implies an O(1) bound for the relaxation time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is (Formula presented.). Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.