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

TL;DR

This paper establishes convergence rates for the density vector in the Curie-Weiss-Potts model using Stein's Method, covering Gaussian and non-Gaussian limits across the entire parameter domain including critical points.

Contribution

It provides explicit convergence rates for the density vector in the Curie-Weiss-Potts model, including at critical points, using Stein's Method of exchangeable pairs.

Findings

Kolmogorov bounds for multivariate normal approximation

Convergence rates at the critical line and extremities

Explicit rates for non-Gaussian limits

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 $\beta\geq 0$ and $h\geq 0$, 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.