Local Central Limit Theorem for Multi-Group Curie-Weiss Models

TL;DR

This paper proves a local central limit theorem for the group magnetisations in a multi-group Curie-Weiss model, extending the known central limit theorem to a more precise probabilistic approximation in the high temperature regime.

Contribution

It introduces a local CLT for multi-group Curie-Weiss models, providing a more detailed probabilistic description of group magnetisations in the high temperature phase.

Findings

Established a local CLT for group magnetisations

Extended the CLT to a finer probabilistic approximation

Applicable in the high temperature regime of the model

Abstract

We define a multi-group version of the mean-field spin model, also called Curie-Weiss model. It is known that, in the high temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is the sum of spins belonging to each group. In this article, we prove a local central limit theorem for the group magnetisations in the high temperature regime.