The statistics of Curie-Weiss models

Let S_n denote the random total magnetization of an n-site Curie-Weiss model, a collection of n (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for large n of the probability distribution of S_n is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (S_n-nm)/n^1-1/2k has a limiting distribution with density proportional to exp[-λ_s^2k/(2k)!], where m is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.