STEIN'S METHOD FOR STATIONARY DISTRIBUTIONS OF MARKOV CHAINS AND APPLICATION TO ISING MODELS

• 成果类型: Article
• 署名作者: Bresler, Guy; Nagaraj, Dheeraj
• 署名单位: Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1479
• 发表日期: 2019
• 页码: 3230-3265
• 关键词: exponential approximation; spectral sparsification; 2nd eigenvalue

摘要

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 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/root 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.