  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 $k$th order moments. Concretely, we show that
$d$-regular Ramanujan graphs approximate the $k$th order moments of the
Curie-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 also
derive some simpler approximation results for functionals of Ising models that
hold only at high enough temperatures.
