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

Kay Kirkpatrick, Tayyab Nawaz

Published 2016-02-10Version 1

We study mean-field classical $N$-vector models, for even 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 a non-normal limit theorem at the critical temperature and central limit theorems away from the critical temperatures. Important special cases of these models are the XY ($N=2$) model of superconductors and Toy ($N=4$) model of the Higgs sector in particle physics.