Arrangement of nearby minima and saddles in the mixed spherical energy landscapes

TL;DR

This paper investigates the geometric arrangement of stationary points in mixed spherical energy landscapes, revealing new structures and challenging previous assumptions about the significance of threshold energy in glassy dynamics.

Contribution

It provides a detailed analysis of the distribution and stability of stationary points in mixed models, highlighting differences from pure models and clarifying the role of threshold energy.

Findings

Rich structure of stationary points in mixed models

Threshold energy's new geometric significance

Invalidates the idea of a continuous manifold of marginal states

Abstract

The mixed spherical models were recently found to violate long-held assumptions about mean-field glassy dynamics. In particular, the threshold energy, where most stationary points are marginal and that in the simpler pure models attracts long-time dynamics, seems to lose significance. Here, we compute the typical distribution of stationary points relative to each other in mixed models with a replica symmetric complexity. We examine the stability of nearby points, accounting for the presence of an isolated eigenvalue in their spectrum due to their proximity. Despite finding rich structure not present in the pure models, we find nothing that distinguishes the points that do attract the dynamics. Instead, we find new geometric significance of the old threshold energy, and invalidate pictures of the arrangement of most marginal inherent states into a continuous manifold.