Dynamical Instantons and Activated Processes in Mean-Field Glass Models

Abstract

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.

arXiv commentv3, conclusions extended and minor revisions

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• Comment on "Explicit Analytical Solution for Random Close Packing in $d=2$ and $d=3$"
• Rare Trajectories in a Prototypical Mean-field Disordered Model: Insights into Landscape and Instantons
• Overlaps between eigenvectors of spiked, correlated random matrices: from matrix PCA to random Gaussian landscapes