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

Eichelsbacher, Peter; Martschink, Bastian

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 252-282.

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 $\beta \ge 0$ et $h\ge 0$, où $\beta$ est l’inverse de la température et $h$ un champ extérieur. Dans ce modèle, la ligne critique $\beta ={\beta }_c\left(h\right)$ 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.

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 $\beta \ge 0$ and $h\ge 0$, where $\beta$ is the inverse temperature and $h$ an exterior field. In this model, the critical line $\beta ={\beta }_c\left(h\right)$ 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.

Classification : 60F05, 82B20, 82B26
Mots clés : Stein’s method, exchangeable pairs, Curie–Weiss–Potts models, critical temperature

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