Stein.s method for stationary distributions of Markov chains and application to Ising models

We develop a new technique, based on Stein.s method, for comparing two stationary distributions of irreducible Markov chains whose update rules are close in a certain sense. 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 k the order moments. Concretely, we show that d-regular Ramanujan graphs approximate the k the order moments of the Curie.Weiss model to within average error k/d... (averaged over size k subsets), independent of graph size. The result applies even in the low-temperature regime; we also derive simpler approximation results for functionals of Ising models that hold only at high temperatures.