Asymptotics of Mean-Field O(N) Models

We study mean-field classical N-vector models, for integers N≥ 2. We use the theory of large deviations and Stein’s 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 Kirkpatrick and Meckes (J Stat Phys 152:54–92, 2013) but with a correction to the critical distribution here], and the Toy (N= 4) model of the Higgs sector in particle physics.