Rigorous results on the bipartite mean-field model (Q2920339)

The paper is devoted to the mathematical properties of the bipartite mean-field model. First, it is computed the exact solution of the thermodynamic limit. A system of Ising spin variables is considered that can be divided into two subsets. Spins interact with each other and with an external field according to the mean-field Hamiltonian. It is exploited a tail estimation on the number of configurations that share the same value of the vector of magnetization. Lower and upper bounds for the partition function that converges to the same value when the number of Ising spin variables aspires to infinity are obtained. This technique has been used by \textit{M. Talagrand} [Spin glasses: A challenge for mathematicians. Cavity and mean field models. Berlin: Springer (2003; Zbl 1033.82002)] in order to compute the thermodynamic limit for the Curie-Weiss model. As a result, the authors obtain a system of mean-field equations in the symmetric regime without an external field which is studied when the interaction coefficients within the two groups are identical. Then, there are analyzed the critical points of the pressure functional associated with the symmetric bipartite mean-field model in the case in which the external field is absent or small. The analysis shows for which values of parameters the model undergoes a phase transition. The results obtained can provide a tool to describe the bipartite non-symmetric mean-field model, in particular they can be used in the statistical approach to socio-economical theories, or the multipartite symmetric mean-field model.