Stein’s method for dependent random variables occuring in statistical mechanics

Abstract

We obtain rates of convergence in limit theorems of partial sums Sn 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 α, a positive number μ, and a positive integer k such that (Sn−nα)/n1−1/2k converges weakly to a random variable with density proportional to exp(−μ|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(−μ|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 βc=1, where the Central Limit Theorem fails. Moreover, we analyze Berry-Esseen bounds as the temperature 1/βn 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.