Ferromagnetic Ising Model on multiregular random graphs

A family of multispecies Ising models on generalized regular random graphs is investigated in the thermodynamic limit. The architecture is specified by class-dependent couplings and magnetic fields. We prove that the magnetizations, neighbours correlations and free energy converge to suitable functions evaluated at the solution of a belief propagation fixed point equation. In absence of magnetic fields, a phase transition is identified and the corresponding critical parameters are determined by the spectral radius of a low-dimensional matrix.