%0 Journal Article %1 bresler2017steins %A Bresler, Guy %A Nagaraj, Dheeraj M. %D 2017 %K stein %T Stein's Method for Stationary Distributions of Markov Chains and Application to Ising Models %U http://arxiv.org/abs/1712.05743 %X 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 $k/d$ (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.

@article{bresler2017steins, abstract = {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.}, added-at = {2017-12-18T14:51:39.000+0100}, author = {Bresler, Guy and Nagaraj, Dheeraj M.}, biburl = {https://www.bibsonomy.org/bibtex/218bcbeb8c608a15f26c446a49c98770d/claired}, description = {Stein's Method for Stationary Distributions of Markov Chains and Application to Ising Models}, interhash = {248088bd6e620f7fe68b82d777d37f15}, intrahash = {18bcbeb8c608a15f26c446a49c98770d}, keywords = {stein}, note = {cite arxiv:1712.05743}, timestamp = {2017-12-18T14:51:39.000+0100}, title = {Stein's Method for Stationary Distributions of Markov Chains and Application to Ising Models}, url = {http://arxiv.org/abs/1712.05743}, year = 2017 }