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October 2019 Stein’s method for stationary distributions of Markov chains and application to Ising models
Guy Bresler, Dheeraj Nagaraj
Ann. Appl. Probab. 29(5): 3230-3265 (October 2019). DOI: 10.1214/19-AAP1479

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 $k$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 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.

Citation

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Guy Bresler. Dheeraj Nagaraj. "Stein’s method for stationary distributions of Markov chains and application to Ising models." Ann. Appl. Probab. 29 (5) 3230 - 3265, October 2019. https://doi.org/10.1214/19-AAP1479

Information

Received: 1 December 2017; Revised: 1 September 2018; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155071
MathSciNet: MR4019887
Digital Object Identifier: 10.1214/19-AAP1479

Subjects:
Primary: 60B10, 60C05, 60F05

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2019
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