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We develop a new technique, based on Stein's method, for comparing twostationary distributions of irreducible Markov Chains whose update rules are`close enough'. We apply this technique to compare Ising models on $d$-regularexpander graphs to the Curie-Weiss model (complete graph) in terms of pairwisecorrelations and more generally $k$th order moments. Concretely, we show that$d$-regular Ramanujan graphs approximate the $k$th order moments of theCurie-Weiss model to within average error $k/\sqrt{d}$ (averaged over the size$k$ subsets). The result applies even in the low-temperature regime; we alsoderive some simpler approximation results for functionals of Ising models thathold only at high enough temperatures.

Stein's Method for Stationary Distributions of Markov Chains and Application to Ising Models