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Path Coupling for Ising and Potts Models Open Access

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We study the convergence behavior of Markov chains for the Ising and Potts models on graphs with n vertices and maximum degree &Delta.; In the case of Glauber dynamics for the Potts model with q colors, we use path coupling to establish rapid mixing (in order nlogn steps) in the high temperature regime where the inverse temperature β is less than log({1+2/Δ}/{1+Δ}). For the Ising model on the complete graph, a well known argument based on path coupling shows that Glauber dynamics has fast mixing. In contrast, for random walk Metropolis chains for the Ising model on the complete graph, we show that a similar approach using path coupling fails to establish a contraction, and therefore does not establish mixing time bounds.

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