Mean-field spin models -- Fluctuation of the magnetization and maximum likelihood estimator

Consider the mean-field spin models where the Gibbs measure of each configuration depends only on its magnetization. Based on the Stein and Laplace methods, we give a new and short proof for the scaling limit theorems with convergence rate for the magnetization in a perturbed model. As an application, we derive the scaling limit theorems for the maximum likelihood estimators (MLEs) in linear models. Remarkably, we characterize the full diagram of fluctuations for the magnetization and MLEs by analyzing the structure of the maximizers of a function associated with the Hamiltonian. For illustration, we apply our results to several well-known mixed spin models, as well as to the annealed Ising model on random regular graphs