  We analyze the energy barriers that allow escapes from a given local minimum
in a mean-field model of glasses. 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 and are associated to
barriers. We find that the lowest barrier, important for activated dynamics at
low temperature, is strictly lower than the "threshold" level above which
saddles proliferate. We characterize how the quenched complexity of barriers,
important for activated process at finite temperature, depends on the energy of
the barrier, 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.
