Asymptotics of mean-field $O(N)$ models

TL;DR

This paper analyzes the asymptotic behavior of the total spin in mean-field classical N-vector models, deriving limit theorems at critical points and away from them using large deviations and Stein's method.

Contribution

It provides new non-normal limit theorems at critical temperatures and clarifies the critical distribution for the Heisenberg model, extending understanding of mean-field N-vector models.

Findings

Derived non-normal limit theorems at critical temperatures.

Established central limit theorems away from criticality.

Corrected the critical distribution for the Heisenberg model.

Abstract

We study mean-field classical $N$-vector models, for integers $N\ge 2$. We use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Important special cases of these models are the XY ($N=2$) model of superconductors, the Heisenberg ($N=3$) model (previously studied in \cite{KM} but with a correction to the critical distribution here), and the Toy ($N=4$) model of the Higgs sector in particle physics.