Role of saddles in mean-field dynamics above the glass transition

Recent numerical developments in the study of glassy systems have shown that it is possible to give a purely geometric interpretation of the dynamic glass transition by considering the properties of unstable saddle points of the energy. Here we further develop this approach in the context of a mean-field model, by analytically studying the properties of the closest saddle point to an equilibrium configuration of the system. We prove that when the glass transition is approached the energy of the closest saddle goes to the threshold energy, defined as the energy level below which the degree of instability of the typical stationary points vanishes. Moreover, we show that the distance between a typical equilibrium configuration and the closest saddle is always very small and that, surprisingly, it is almost independent of the temperature.