Computing the number of metastable states in infinite-range models

TL;DR

This paper reviews recent results on calculating the large number of metastable states in infinite-range disordered models, highlighting the role of supersymmetry and its breaking, especially in the context of spin glasses.

Contribution

It provides a detailed analysis of the computation of metastable states in infinite-range models and elucidates the significance of supersymmetry in this context.

Findings

Most stationary points in the Sherrington-Kirkpatrick model are saddles.

Supersymmetry is crucial in understanding the landscape of stationary points.

Supersymmetry breaking has significant physical implications.

Abstract

In these notes I will review the results that have been obtained in these last years on the computation of the number of metastable states in infinite-range models of disordered systems. This is a particular case of the problem of computing the exponentially large number of stationary points of a random function. Quite surprisingly supersymmetry plays a crucial role in this problem. A careful analysis of the physical implication of supersymmetry and of supersymmetry breaking will be presented: the most spectacular one is that in the Sherrington-Kirkpatrick model for spin glasses most of the stationary points are saddles, as predicted long time ago.