Critical behavior of mean-field XY and related models

We discuss spin models on complete graphs in the mean-field (infinite-vertex) limit, especially the classical XY model, the Toy model of the Higgs sector, and related generalizations. We present a number of results coming from the theory of large deviations and Steins method, in particular, Cramer and Sanov-type results, limit theorems with rates of convergence, and phase transition behavior for these models.

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

We study mean-field classical $N$-vector models, for integers $Nge 2$. We use the theory of large deviations and Steins 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.