Existence of two-step replica symmetry breaking for the spherical mixed p-spin glass at zero temperature

TL;DR

This paper demonstrates the existence of two-step replica symmetry breaking in spherical mixed p-spin glasses at zero temperature, revealing new structural insights and answering longstanding classification questions.

Contribution

It provides the first examples of 2-RSB models at zero temperature with a purely atomic Parisi measure having three support points.

Findings

Parisi measure is purely atomic with three points

Implications for the topology of the energy landscape

Negative answer to a 2011 classification question

Abstract

We provide the first examples of two-step replica symmetry breaking (2-RSB) models for the spherical mixed p-spin glass at zero temperature. Precisely, we show that for a certain class of mixtures, the Parisi measure at zero temperature is purely atomic and has exactly three distinct points in its support. We then derive a few consequences for the topology of the random landscape in these cases. Our main result also provides a negative answer to a question raised in 2011 by Auffinger and Ben Arous about the classification of pure-like and full mixture models.