The Swendsen-Wang Dynamics on Trees
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The Swendsen-Wang algorithm is a sophisticated, widely-used Markov chain for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models with inverse temperature β>0. Due to the global nature of the updates of the Swendsen-Wang algorithm, the underlying Markov chain often mixes rapidly in the low temperature region (large β) where long-range correlations impair the convergence rate of local chains such as the Glauber dynamics. With few exceptions, tight bounds on the convergence rate of the Swendsen-Wang algorithm only hold for high temperature regions (small β) corresponding to the uniqueness or the decay of correlations region. We present tight bounds on the convergence rate of the Swendsen-Wang algorithm for the complete d-ary tree that extend to the non-uniqueness region. Specifically, we show that a spatial mixing property known as the Variance Mixing condition introduced by Martinelli et al. (2004) implies constant spectral gap of the Swendsen-Wang dynamics. This implies that the relaxation time (i.e., the inverse of the spectral gap) is O(1) for all boundary conditions in the uniqueness region or when β<β_1 where β_1 exceeds the uniqueness threshold for the Ising model and for the q-state Potts model when q is small with respect to d. In addition, we prove O(1) relaxation time for all β for the monochromatic boundary condition. Our proof introduces a novel spectral view of the Variance Mixing condition inspired by several recent rapid mixing results on high-dimensional expanders.
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