Stein’s method for stationary distributions of Markov chains and application to Ising models
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1712.05743.pdf
Description
Accepted version
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502.98 KB
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Author(s)
Bresler, Guy
Nagaraj, Dheeraj
Date Issued
October 2019
Journal
Annals of Applied Probability
Publisher
Institute of Mathematical Statistics
Citation
Bresler, Guy and Dheeraj Nagaraj. "Stein’s method for stationary distributions of Markov chains and application to Ising models." Annals of Applied Probability 29, 5 (October 2019): 3230 - 3265 © 2019 Institute of Mathematical Statistics
Version
Author's final manuscript
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 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/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.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1214/19-aap1479
