  We focus on the energy landscape of a simple mean-field model of glasses and
analyze activated barrier-crossing by combining the Kac-Rice method for
high-dimensional Gaussian landscapes with dynamical field theory. In
particular, we consider Langevin dynamics at low temperature in the energy
landscape of the pure spherical $p$-spin model. We select as initial condition
for the dynamics one of the many unstable index-1 saddles in the vicinity of a
reference local minimum. We show that the associated dynamical mean-field
equations admit two solutions: one corresponds to falling back to the original
reference minimum, and the other to reaching a new minimum past the barrier. By
varying the saddle we scan and characterize the properties of such minima
reachable by activated barrier-crossing. Finally, using time-reversal
transformations, we construct the two-point function dynamical instanton of the
corresponding activated process.
