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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.
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 Sa
Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining these with
We construct the exact partition function of the Potts model on a complete graph subject to external fields with linear and nematic type couplings. The partition function is obtained as a solution to a linear diffusion equation and the free energy, i
We consider the $n$-component $|varphi|^4$ lattice spin model ($n ge 1$) and the weakly self-avoiding walk ($n=0$) on $mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or
We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar ro