Free energy and fluctuations of generalized bipartite Sherrington-Kirkpatrick models and generalized Hopfield models

We consider the free energy of generalized bipartite Sherrington-Kirkpatrick models in the high temperature regime, with one spin normalized and the other either normalized or i.i.d. We derive a unified formula for the limiting free energy in both settings. We also prove that the fluctuations of the free energy converge to a Gaussian distribution under mild assumptions. The proof is based on a bipartite multigraph expansion, which also gives free energy results for generalized Hopfield models and log likelihood ratio of spiked rectangular models.