Dynamics for Spherical Spin Glasses: Disorder Dependent Initial Conditions

We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed p -spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure p -spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature).