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 $k/d$ (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.
Description
Stein's Method for Stationary Distributions of Markov Chains and
Application to Ising Models
%0 Journal Article
%1 bresler2017steins
%A Bresler, Guy
%A Nagaraj, Dheeraj M.
%D 2017
%K stein
%T Stein's Method for Stationary Distributions of Markov Chains and
Application to Ising Models
%U http://arxiv.org/abs/1712.05743
%X 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 $k/d$ (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.
@article{bresler2017steins,
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.},
added-at = {2017-12-18T14:51:39.000+0100},
author = {Bresler, Guy and Nagaraj, Dheeraj M.},
biburl = {https://www.bibsonomy.org/bibtex/218bcbeb8c608a15f26c446a49c98770d/claired},
description = {Stein's Method for Stationary Distributions of Markov Chains and
Application to Ising Models},
interhash = {248088bd6e620f7fe68b82d777d37f15},
intrahash = {18bcbeb8c608a15f26c446a49c98770d},
keywords = {stein},
note = {cite arxiv:1712.05743},
timestamp = {2017-12-18T14:51:39.000+0100},
title = {Stein's Method for Stationary Distributions of Markov Chains and
Application to Ising Models},
url = {http://arxiv.org/abs/1712.05743},
year = 2017
}