Gaussian and non-Gaussian fluctuations in the two- and five-spin mean-field ising model

Abstract

Abstract We investigate a mean-field Ising model with simultaneous two-spin and five-spin interactions. Using a variational principle for the Gibbs free energy, we provide a complete characterization of the phase diagram and identify the coexistence curve separating polarized and unpolarized phases. We prove law of large numbers and central limit theorems for the magnetization away from criticality, establish non-Gaussian fluctuation limits at the critical point, and compute the associated critical exponents. In particular, we show that the addition of quintic interaction terms induces a discontinuous first-order phase transition, in contrast to the second-order behaviour of the classical Curie–Weiss model. Our results extend the framework of mean-field limit theorems for higher-order Ising models and highlight the role of five-body couplings in shaping collective phenomena.