Asymptotics of the mean-field Heisenberg model

TL;DR

This paper analyzes the asymptotic behavior of the total spin in the mean-field classical Heisenberg model, revealing phase transitions and limit theorems through large deviations and Stein's method.

Contribution

It provides detailed large deviations principles and limit theorems for the total spin, including at the critical temperature, using Stein's method.

Findings

Established large deviations principles for total spin and empirical distribution.

Proved central limit theorems in sub- and supercritical phases.

Derived a nonnormal limit theorem at the critical temperature.

Abstract

We consider the mean-field classical Heisenberg model and obtain detailed information about the total spin of the system by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cramer- and Sanov-type large deviations principles for the total spin and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the total spin throughout the phase transition using Stein's method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature.