Stein’s Method and a Cubic Mean-Field Model

Abstract

In this paper, we study a mean-field spin model with three- and two-body interactions. In a recent paper (Ann Henri Poincaré, 2024) by Contucci, Mingione and Osabutey, the equilibrium measure for large volumes was shown to have three pure states, two with opposite magnetization and an unpolarized one with zero magnetization, merging at the critical point. The authors proved a central limit theorem for the suitably rescaled magnetization. The aim of our paper is presenting a prove of a central limit theorem for the rescaled magnetization applying the exchangeable pair approach due to Stein. Moreover we prove (non-uniform) Berry–Esseen bounds, a concentration inequality, Cramér-type moderate deviations and a moderate deviations principle for the suitably rescaled magnetization. Interestingly we analyze Berry–Esseen bounds in case the model-parameters \((K_n,J_n)\) converge to the critical point (0, 1) on lines with different slopes and with a certain speed, and obtain new limiting distributions and thresholds for the speed of convergence.