The Annals of Applied Probability

Stein’s method for stationary distributions of Markov chains and application to Ising models

Guy Bresler and Dheeraj Nagaraj

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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.

Article information

Ann. Appl. Probab., Volume 29, Number 5 (2019), 3230-3265.

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

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Mathematical Reviews number (MathSciNet)

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 60B10: Convergence of probability measures

Ising model Stein’s method graph sparsification Curie–Weiss


Bresler, Guy; Nagaraj, Dheeraj. Stein’s method for stationary distributions of Markov chains and application to Ising models. Ann. Appl. Probab. 29 (2019), no. 5, 3230--3265. doi:10.1214/19-AAP1479.

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