Complexity of the Sherrington-Kirkpatrick Model in the Annealed Approximation

TL;DR

This paper critically analyzes the complexity of the Sherrington-Kirkpatrick model at the annealed level, examining supersymmetry invariance and saddle points, and discusses inconsistencies in previous solutions.

Contribution

It provides a detailed critical analysis of the annealed complexity functional, highlighting invariance properties and comparing different saddle point solutions.

Findings

Complexity functional is invariant under supersymmetry.

Two saddle points examined: one supersymmetric, one breaking supersymmetry.

Identifies inconsistencies in previous solutions.

Abstract

A careful critical analysis of the complexity, at the annealed level, of the Sherrington-Kirkpatrick model has been performed. The complexity functional is proved to be always invariant under the Becchi-Rouet-Stora-Tyutin supersymmetry, disregarding the formulation used to define it. We consider two different saddle points of such functional, one satisfying the supersymmetry [A. Cavagna {\it et al.}, J. Phys. A {\bf 36} (2003) 1175] and the other one breaking it [A.J. Bray and M.A. Moore, J. Phys. C {\bf 13} (1980) L469]. We review the previews studies on the subject, linking different perspectives and pointing out some inadequacies and even inconsistencies in both solutions.