Exactly solvable spin-glass models with ferromagnetic couplings: the spherical multi-$p$-spin model in a self-induced field

We report some results on the quenched disordered Spherical multi-$p$-Spin Model in presence of ferromagnetic couplings. In particular, we present the phase diagrams of some representative cases that schematically describe, in the mean-field approximation, the behavior of most known transitions in glassy materials, including dynamic arrest in super-cooled liquids, amorphous-amorphous transitions and spin-glass transitions. A simplified notation is introduced in order to compute systems properties in terms of an effective, self-induced, field encoding the whole ferromagnetic information.