Beyond pairwise interaction: the cubic mean-field Ising model

The system under consideration comprises Ising spins, a homogeneous magnetic field, and a constant two-spin interaction, augmented by a constant three-spin interaction term. Employing large deviation methods, this thesis explores the thermodynamic properties of the system. The equilibrium measure for large volumes exhibits three pure states when the magnetic field is absent, corresponding to phases of the system, encompassing two with opposite magnetisation and an unpolarized state with zero magnetisation, converging at the critical point. Analysis of the system with three-spin interaction and an external field reveals two coexistence curves, indicative of two distinct second-order phase transitions contingent on the interaction parameter domain. The critical exponents of the magnetic order parameter are calculated in all directions of the phase space and show their agreement with the mean-field universality class. While the Central Limit Theorem holds for a suitably rescaled magnetization, its violation of the typical quartic behaviour appears at the critical point. Furthermore, the thesis solves the inverse problem for the model, reconstructing the system's free parameters from configuration data generated according to the model's distribution. The robustness of this inversion procedure is tested within both unique solution regions and regions with multiple thermodynamic phases. Lastly, a multi-populated system with three-spin interaction is solved, serving as a paradigm for modelling complex systems comprising human and AI agents. Notably, the study underscores that for suitable values of the interaction parameters, arbitrarily small values of the relative size of the AI agents may trigger dramatic changes in the system.