Phase diagram and fluctuations in the cubic mean field Ising model

We study the cubic mean field Ising model with three body and two body interactions and an external field. Our main result is a complete classification of the global maximizers of the variational pressure in the full three-parameter region $(K,J,h)\in (0,\mathrm{\infty })\times {\mathbb{R}}^2$. We identify a coexistence surface with two nondegenerate global maximizers and a critical curve with a unique quartically degenerate maximizer; elsewhere the maximizer is unique and nondegenerate. This extends the phase diagrams previously obtained on two-parameter subfamilies \cite{CMO24,O24}. As applications of the general results in \cite{CR}, we give explicit magnetization fluctuation theorems and asymptotic distributions for one-parameter maximum likelihood estimators in the resulting regimes.