Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Stein’s method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process

Larry Goldstein and Nathakhun Wiroonsri

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We provide nonasymptotic $L^{1}$ bounds to the normal for four well-known models in statistical physics and particle systems in $\mathbb{Z}^{d}$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^{1}$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for the total number of points in a finite region belonging to an infinite cluster in dimensions $d\ge2$, in the voter model for the occupation time of the origin in dimensions $d\ge7$, and for finite time integrals of nonconstant increasing cylindrical functions evaluated on the one dimensional supercritical contact process started in its unique invariant distribution.

The tool developed for these purposes is a version of Stein’s method adapted to positively associated random variables. In one dimension, letting $\mathbf{{\xi}=(\xi_{1},\ldots,\xi_{m})}$ be a positively associated mean zero random vector with components that obey the bound $\vert \xi_{i}\vert \le B,i=1,\ldots,m$, and whose sum $W=\sum_{i=1}^{m}\xi_{i}$ has variance 1, it holds that \begin{eqnarray*}d_{1}(\mathcal{L}(W),\mathcal{L}(Z))\le5B+\sqrt{\frac{8}{\pi}}\sum_{i\neq j}\mathbb{E}[\xi_{i}\xi_{j}],\end{eqnarray*} where $Z$ has the standard normal distribution and $d_{1}(\cdot,\cdot)$ is the $L^{1}$ metric. Our methods apply in the multidimensional case with the $L^{1}$ metric replaced by a smooth function metric.


Nous obtenons des bornes $L^{1}$ non asymptotiques montrant l’approximation gaussienne pour quatre modèles classiques de physique statistique et de systèmes de particules dans $\mathbb{Z}^{d}$ : le modèle d’Ising ferromagnétique au plus proche voisin, la percolation par arêtes sur-critique, le modèle du votant et le processus de contact. Pour le modèle d’Ising, nous obtenons une borne en distance $L^{1}$ entre la magnétisation totale et la loi normale, soit à toute température lorsque le paramètre de moment magnétique est non nul, soit, dans la phase sous-critique, si ce paramètre est nul. Pour le modèle de percolation, nous obtenons une telle borne pour le nombre total de points dans une région finie qui appartiennent à un cluster infini, en dimensions $d\geq2$. Pour le modèle du votant nous considérons le temps d’occupation à l’origine en dimensions $d\geq7$, et pour le processus de contact, nous nous intéressons à des intégrales temporelles à horizon fini de fonctions cylindriques croissantes, évaluées en le processus de contact unidimensionnel sous l’unique mesure invariante.

L’outil que nous développons à cette fin est une version de la méthode de Stein adaptée à des variables aléatoires associées positivement. En dimension $1$, si $\mathbf{{\xi}=(\xi_{1},\ldots,\xi_{m})}$ désigne un vecteur aléatoire centré et positivement associé dont les composantes satisfont la borne $\vert \xi_{i}\vert \le B,i=1,\ldots,m$, et dont la somme $W=\sum_{i=1}^{m}\xi_{i}$ est de variance 1, alors \begin{eqnarray*}d_{1}(\mathcal{L}(W),\mathcal{L}(Z))\le5B+\sqrt{\frac{8}{\pi}}\sum_{i\neq j}\mathbb{E}[\xi_{i}\xi_{j}]\end{eqnarray*} où $Z$ est une variable aléatoire gaussienne centrée réduite et $d_{1}(\cdot,\cdot)$ est la distance $L^{1}$. Nos méthodes s’appliquent au cas multidimensionnel si la distance $L^{1}$ est remplacée par une distance plus faible.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 385-421.

Received: 1 April 2016
Revised: 17 October 2016
Accepted: 18 November 2016
First available in Project Euclid: 19 February 2018

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 82B30: Statistical thermodynamics [See also 80-XX] 60G60: Random fields

Random fields Block dependence Correlation inequality Positive dependence


Goldstein, Larry; Wiroonsri, Nathakhun. Stein’s method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 385--421. doi:10.1214/16-AIHP808.

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