Quenched Computation of the Complexity of the Sherrington-Kirkpatrick Model

TL;DR

This paper presents a method to compute the complexity of the Sherrington-Kirkpatrick model using a modified supersymmetric Ansatz, revealing inconsistencies at non-equilibrium levels and discussing the challenge of defining physically meaningful solutions.

Contribution

Introduces a supersymmetry-based approach to compute the quenched complexity, highlighting its limitations and discussing the construction of physically relevant solutions.

Findings

Complexity is inconsistent outside the equilibrium state.

The Legendre transform relates complexity to quenched free energy.

Supersymmetry invariance leads to limitations in describing metastable states.

Abstract

The quenched computation of the complexity in the Sherrington-Kirkpatrick model is presented. A modified Full Replica Symmetry Breaking Ansatz is introduced in order to study the complexity dependence on the free energy. Such an Ansatz corresponds to require Becchi-Rouet-Stora-Tyutin supersymmetry. The complexity computed this way is the Legendre transform of the free energy averaged over the quenched disorder. The stability analysis shows that this complexity is inconsistent at any free energy level but the equilibirum one. The further problem of building a physically well defined solution not invariant under supersymmetry and predicting an extensive number of metastable states is also discussed.