Two Groups in a Curie-Weiss Model with Heterogeneous Coupling

TL;DR

This paper analyzes a two-group Curie-Weiss model with heterogeneous couplings, establishing laws of large numbers, a central limit theorem in the high temperature regime, and non-Gaussian limits at criticality, revealing complex phase behaviors.

Contribution

It introduces a two-group Curie-Weiss model with different intra- and inter-group couplings and derives new probabilistic limit theorems across temperature regimes.

Findings

Bivariate laws of large numbers for group magnetizations

Central limit theorem in the high temperature regime

Non-Gaussian limit distribution at criticality

Abstract

We discuss a Curie-Weiss model with two groups with different coupling constants within and between groups. For the total magnetisations in each group, we show bivariate laws of large numbers and a central limit theorem which is valid in the high temperature regime. In the critical regime, the total magnetisation normalised by $N^{3/4}$ converges to a non-trivial distribution which is not Gaussian, just as in the single-group Curie-Weiss model. Finally, we prove a kind of a `law of large numbers' in the low temperature regime, more precisely we prove that the empirical magnetisation converges in distribution to a mixture of two Dirac measures.