Stein's method for dependent random variables occurring in Statistical Mechanics

By Peter Eichelsbacher, Matthias Löwe. 2009

52 citations recorded by the source corpus.

We obtain rates of convergence in limit theorems of partial sums $S_n$ for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the Curie-Weiss models. Under appropriate assumptions there exists a real number $\alpha$, a positive real number $\mu$, and a positive integer $k$ such that $(S_n- n \alpha)/n^{1 - 1/2k}$ converges weakly to a random variable with density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. We develop Stein's method for exchangeable pairs for a rich class of distributional approximations including the Gaussian distributions as well as the non-Gaussian limit distributions with density proportional to $\exp(-\mu |x|^{2k} /(2k)!)$. Our results include the optimal Berry-Esseen rate in the Central Limit Theorem for the total magnetization in the classical Curie-Weiss model, for high temperatures as well as at the critical temperature $\beta_c=1$, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature $1/ \beta_n$ converges to one and obtain a threshold for the speed of this convergence. Single spin distributions satisfying the Griffiths-Hurst-Sherman (GHS) inequality like models of liquid helium or continuous Curie-Weiss models are considered.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/91b39717403a621c402ce46b511907db5f029ccc

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Citations in the corpus

Stein's Method of Exchangeable Pairs with Application to the Curie-Weiss Model (2009, background): “Incidentally, after this manuscript was submitted, it was brought to our attention that an article by Eichelsbacher and Löwe (2009) was in preparation, where the same result (Theorem 2.1) is proved, along the same lines as our proof.”
Stein's method for comparison of univariate distributions (2014, methodology): “It can also be tailored for specific problems such as preferential attachment graphs [80], the Curie-Weiss model [21], and other models from statistical mechanics [32, 33].”
A short survey of Stein's method (2014, methodology): “Stein’s method for a mixture of two normal distributions, with an application to spin glasses, appeared in my 2010 paper [23], while another non-normal distribution arising at the critical temperature of the Curie-Weiss model of ferromagnets was tackled in my joint paper with Shao [30] in 2011 and i”
Applications of Stein's method for concentration inequalities (2009, result): “This limit theorem was first proved by Simon and Griffiths [35] and error bounds were obtained recently [13, 16].”