Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models

TL;DR

This paper characterizes the ground state energy of spherical mixed p-spin models through a variational formula, extending the Parisi formula, and explores disorder chaos and fluctuation phenomena at zero temperature.

Contribution

It provides a variational characterization of the ground state energy, explicit formulas across different replica symmetry phases, and new insights into disorder chaos and fluctuations.

Findings

Ground state energy is given by an infimum of a variational problem.

Disorder chaos occurs in the absence of external field.

Ground state energy superconcentrates without external field; obeys CLT with external field.

Abstract

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.