On the energy landscape of spherical spin glasses

TL;DR

This paper explores the complex energy landscape of spherical spin glasses, revealing how the structure of near maxima relates to the Parisi measure and differing behaviors between one-step and full replica symmetry breaking models.

Contribution

It establishes a connection between the distance of near maxima and the Parisi measure, and characterizes the geometry of configurations in models with different replica symmetry breaking.

Findings

Near maxima are asymptotically orthogonal in one-step RSB models.

Full RSB models exhibit a continuum of asymptotic distances near maximum energy.

Provides algebraic relations characterizing Parisi measures at zero temperature.

Abstract

We investigate the energy landscape of the spherical mixed even p-spin model near its maximum energy. We relate the distance between pairs of near maxima to the support of the Parisi measure at zero temperature. We then provide an algebraic relation that characterizes one-step replica symmetric breaking Parisi measures. For these measures, we show that any two nonparallel spin configurations around the maximum energy are asymptotically orthogonal to each other. In sharp contrast, we study models with full replica symmetry breaking and show that all possible values of the asymptotic distance are attained near the maximum energy.