Stein's Method for Stationary Distributions of Markov Chains and Application to Ising Models
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We develop a new technique, based on Stein's method, for comparing two stationary distributions of irreducible Markov Chains whose update rules are `close enough'. We apply this technique to compare Ising models on d-regular expander graphs to the Curie-Weiss model (complete graph) in terms of pairwise correlations and more generally kth order moments. Concretely, we show that d-regular Ramanujan graphs approximate the kth order moments of the Curie-Weiss model to within average error k/√(d) (averaged over the size k subsets). The result applies even in the low-temperature regime; we also derive some simpler approximation results for functionals of Ising models that hold only at high enough temperatures.
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