p-spin-interaction spin-glass models: Connections with the structural glass problem

The static and the dynamical theories for the mean-field p-spin (p>2) interaction spin-glass model are studied. A broken-replica-symmetric equilibrium solution leads to a glass transition at a temperature $T_g^{\mathcal{’}}$ where the Edwards-Anderson order parameter is discontinuous but where there is no latent heat and there is a discontinuous specific heat. The dynamical theory leads to a continuous slowing down and predicts a glass transition at $T_g$>$T_g^{\mathcal{’}}$. A reinterpretation of the equilibrium solution allows us to relate the dynamical transition to the equilibrium theory. The mathematical structure of the mean-field dynamical theory is closely related to certain recent dynamical theories of the structural glass transition.