Decay of correlations and non-local Markov chains
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In this talk we consider Markov chains for spin systems on the integer lattice graph Z^d. It has been well known since pioneering work from the early 1990’s that a certain decay of correlation property, known as strong spatial mixing (SSM), is a necessary and sufficient condition for fast mixing of the Gibbs sampler, where the state of a single vertex is updated in each step. In practice, non-local Markov chains are particularly popular from their potentially exponential speed-up, but these processes have largely resisted analysis. In this talk, we consider the effects of SSM on the rate of convergence to stationary of non-local Markov chains. We show that SSM implies fast mixing of several standard non-local chains, including general blocks dynamics, systematic scan dynamics and the Swendsen-Wang dynamics for the Ising/Potts model. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.
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