Limit theorems for the non-convex multispecies Curie-Weiss model

TL;DR

This paper investigates the thermodynamic behavior of a generalized non-convex multispecies Curie-Weiss model, deriving pressure formulas and establishing a central limit theorem for magnetization fluctuations under various conditions.

Contribution

It introduces a method to compute the pressure for the model with generic priors and analyzes the fluctuation behavior of magnetizations for Ising spins, including CLT results.

Findings

Pressure computed via interpolation techniques for generic priors.

CLT established for species magnetizations, with convergence to normal distributions.

Dependence of fluctuation limits on the relative sizes of species.

Abstract

We study the thermodynamic properties of the generalized non-convex multispecies Curie-Weiss model, where interactions among different types of particles (forming the species) are encoded in a generic matrix. For spins with a generic prior distribution, we compute the pressure in the thermodynamic limit using simple interpolation techniques. For Ising spins, we further analyze the fluctuations of the magnetization in the thermodynamic limit under the Boltzmann-Gibbs measure. It is shown that a central limit theorem holds for a rescaled and centered vector of species magnetizations, which converges to either a centered or non-centered multivariate normal distribution, depending on the rate of convergence of the relative sizes of the species.