Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model

We study the nonequilibrium relaxation of the spherical spin-glass model with p-spin interactions in the N→∞ limit. We analytically solve the asymptotics of the magnetization and the correlation and response functions for long but finite times. Even in the thermodynamic limit the system exhibits ‘‘weak’’ (as well as ‘‘true’’) ergodicity breaking and aging effects. We determine a functional Parisi-like order parameter ${\mathit{P}}_{\mathit{d}}$(q) which plays a similar role for the dynamics to that played by the usual function for the statics.