Activated processes and Inherent Structure dynamics of finite-size mean-field models for glasses

We investigate the inherent structure (IS) dynamics of mean-field {\it finite-size} spin-glass models whose high-temperature dynamics is described in the thermodynamic limit by the schematic Mode Coupling Theory for super-cooled liquids. Near the threshold energy the dynamics is ruled by activated processes which induce a logarithmic slow relaxation. We show the presence of aging in both the IS correlation and integrated response functions and check the validity of the one-step replica symmetry breaking scenario in the presence of activated processes. Our work shows: 1) The violation of the fluctuation-dissipation theorem is given by the configurational entropy, 2) The intermediate time regime ($\log (t)\sim N$) in mean-field theory automatically includes activated processes opening the way to analytically investigate activated processes by computing corrections beyond mean-field.