Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Gaussian fluctuations for the classical XY model

Charles M. Newman and Wei Wu

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We study the classical XY model in bounded domains of $\mathbb{Z}^{d}$ with Dirichlet boundary conditions. We prove that when the temperature goes to zero faster than a certain rate as the lattice spacing goes to zero, the fluctuation field converges to a Gaussian white noise. This and related results also apply to a large class of gradient field models.


Nous étudions le modèle XY classique dans un domaine borné de $\mathbb{Z}^{d}$ avec condition de Dirichlet au bord. Nous prouvons que quand la température tend vers 0 suffisamment vite avec le pas du graphe, le champ des fluctuations converge vers le bruit blanc Gaussien. Ce résultat ainsi que les résultats associés s’appliquent aussi à une classe large de modèles de champs gradients.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 1759-1777.

Received: 29 December 2016
Accepted: 13 July 2017
First available in Project Euclid: 18 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
Secondary: 60F17: Functional limit theorems; invariance principles 60G60: Random fields

XY model Spin-wave approximation Gaussian free field Gradient field models Random walk representation Central limit theorem


Newman, Charles M.; Wu, Wei. Gaussian fluctuations for the classical XY model. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 1759--1777. doi:10.1214/17-AIHP854.

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