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Stein's Method for Stationary Distributions of Markov Chains and Application to Ising Models
Probability · Submitted 15 Dec 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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).
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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'. | contribution 0.89 | 0.96 | 0.14 | asserted 1.00 |
| s1 | 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. | contribution 0.92 | 0.93 | 0.15 | asserted 0.97 |
| s2 | 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). | result 1.00 | 0.95 | 0.60 | asserted 1.00 |
| s3 | 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. | result 0.69 | 0.94 | 0.08 | asserted 1.00 |
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The paper
Original 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.
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How the literature uses it
3 arXiv papers cite this in 3 sentences
Papers citing it · 1 sentence without a confident label
Compares with it 1 paper
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It was shown in [ref] that pairwise correlations, and more generally $k$th-order moments, of the Curie-Weiss model can be well approximated on average by expander graphs, yet the result above holds even when the underlying graph is an expander.
Optimal Single Sample Tests for Structured versus Unstructured Network Data · Introduction · 0.88
Cites it as background 1 paper
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The notions of statistical distances in relation with Markov chains, in particular with respect to their stationary distributions have also been explored in [ref] .
Statistical Windows in Testing for the Initial Distribution of a Reversible Markov Chain · Introduction · 0.99
What it cites · 1 sentence without a confident label
Compares with 1 paper
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The same result as stated in Theorem [ref], with a similar proof, was discovered independently in [ref] .
Approximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphs · Introduction · 1.00
Cites as background 1 paper
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The problem of testing between dense and sparse Ising models is studied in [ref] .
Optimal Single Sample Tests for Structured versus Unstructured Network Data · Introduction · 0.99
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