Complexity of energy barriers in mean-field glassy systems

We analyze the energy barriers that allow escapes from a given local minimum in a complex high-dimensional landscape. We perform this study by using the Kac-Rice method and computing the typical number of critical points of the energy function at a given distance from the minimum. We analyze their Hessian in terms of random matrix theory and show that for a certain regime of energies and distances critical points are index-one saddles, or transition states, and are associated to barriers. We find that the transition state of lowest energy, important for the activated dynamics at low temperature, is strictly below the "threshold" level above which saddles proliferate. We characterize how the quenched complexity of transition states, important for the activated processes at finite temperature, depends on the energy of the state, the energy of the initial minimum, and the distance between them. The overall picture gained from this study is expected to hold generically for mean-field models of the glass transition.