Stein's Method for Stationary Distributions of Markov Chains and Application to Ising Models

TL;DR

This paper introduces a Stein's method-based technique to compare stationary distributions of Markov chains, specifically applying it to approximate Ising models on regular graphs by the Curie-Weiss model, with quantifiable error bounds.

Contribution

The paper develops a novel Stein's method approach for comparing stationary distributions of Markov chains and applies it to approximate complex Ising models with simpler models on expander graphs.

Findings

d-regular Ramanujan graphs approximate Curie-Weiss moments within error k/√d

Approximation holds even in low-temperature regimes

Simpler high-temperature functional approximations derived

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.