Phase Diagram of Coupled Glassy Systems: A Mean-Field Study

In the example of the spherical $p$-spin model, we study the phase diagram of glassy systems in the presence of an attractive coupling with a quenched configuration. We find competition among two phases, separated by a coexistence line terminating in a critical point, as in ordinary first-order phase transitions. We argue that these results are not an artifact of the mean-field approximation, and may be observed in numerical simulations of realistic glassy models.