Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On rates of convergence in the Curie–Weiss–Potts model with an external field

Peter Eichelsbacher and Bastian Martschink

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Abstract

In the present paper we obtain rates of convergence for the limit theorems of the density vector in the Curie–Weiss–Potts model via Stein’s Method of exchangeable pairs. Our results include Kolmogorov bounds for multivariate normal approximation in the whole domain $\beta\geq0$ and $h\geq0$, where $\beta$ is the inverse temperature and $h$ an exterior field. In this model, the critical line $\beta=\beta_{c}(h)$ is explicitly known and corresponds to a first order transition. We include rates of convergence for non-Gaussian approximations at the extremity of the critical line of the model.

Résumé

Dans cet article, nous obtenons des taux de convergence pour les vecteurs de densité dans le modèle de Curie–Weiss–Potts via la méthode de Stein des paires échangeables. Nos résultats incluent des bornes de Kolmogorov pour l’approximation normale multivariée dans tout le domaine $\beta\geq0$ et $h\geq0$, où $\beta$ est l’inverse de la température et $h$ un champ extérieur. Dans ce modèle, la ligne critique $\beta=\beta_{c}(h)$ est explicitement connue et correspond à une transition du premier ordre. Nous incluons des taux de convergence pour des approximations non-gaussiennes au bord de la ligne critique du modèle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 1 (2015), 252-282.

Dates
First available in Project Euclid: 14 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1421244405

Digital Object Identifier
doi:10.1214/14-AIHP599

Mathematical Reviews number (MathSciNet)
MR3300970

Zentralblatt MATH identifier
1321.60038

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general)

Keywords
Stein’s method Exchangeable pairs Curie–Weiss–Potts models Critical temperature

Citation

Eichelsbacher, Peter; Martschink, Bastian. On rates of convergence in the Curie–Weiss–Potts model with an external field. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 1, 252--282. doi:10.1214/14-AIHP599. https://projecteuclid.org/euclid.aihp/1421244405


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