Rescaled Magnetization for Critical Bipartite Mean-Fields Models

We consider a bipartite generalization of the Curie-Weiss model in a critical regime. In order to study the asymptotic behavior of the random vector of the total magnetization we apply the change of variables that diagonalizes the Hessian matrix of the pressure functional associated to the model. We obtain a new vector that, suitably rescaled, weakly converges to the product of a Gaussian distribution and a distribution proportional to $\exp(-\xi x^{4})$, where the positive constant $\xi$ can be computed from the pressure functional.