Limit theorems for the non-convex multispecies Curie-Weiss model
Francesco Camilli, Emanuele Mingione, Godwin Osabutey
TL;DR
This work analyzes a generalized multispecies Curie–Weiss model with arbitrary spin priors and a non-definite interaction matrix. It derives a variational representation for the thermodynamic pressure in the large-$N$ limit via interpolation techniques and establishes central limit theorems for the magnetization vector in the Ising case. The CLTs reveal either centered or non-centered Gaussian fluctuations, depending on the convergence rate of species proportions and the multiplicity of global maximizers of an auxiliary function, with conditional results when multiple maxima prevail. The results broaden the mean-field analysis to non-convex interactions and continuous spin spaces, enabling applications to complex networks, social dynamics, and meta-magnet-like systems where both cooperative and antagonistic interactions coexist.
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.