Dynamical Instantons and Activated Processes in Mean-Field Glass Models

TL;DR

This paper investigates activated barrier-crossing in mean-field glass models by combining high-dimensional landscape analysis with dynamical theory, revealing how systems transition between minima via instantons.

Contribution

It introduces a novel approach combining Kac-Rice and dynamical field theory to analyze activated processes and instantons in mean-field glass models.

Findings

Identification of two dynamical solutions: return to original minimum or transition to a new minimum.

Characterization of properties of minima reachable through barrier-crossing.

Construction of the dynamical instanton describing the activated process.

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.