Complexity of energy barriers in mean-field glassy systems

TL;DR

This paper investigates the energy barriers in mean-field glassy systems, revealing how critical points and saddles influence activated dynamics and the glass transition.

Contribution

It introduces a novel analysis of energy barriers using the Kac-Rice method and random matrix theory, providing new insights into the structure of critical points in glassy landscapes.

Findings

Lowest barriers are below the threshold level where saddles proliferate.

Critical points at certain energies and distances are index-one saddles.

The complexity of barriers depends on barrier and minimum energies, and their separation.

Abstract

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.