Index Distribution of Random Matrices with an Application to Disordered Systems

TL;DR

This paper calculates the distribution of negative eigenvalues in Gaussian random matrices using the replica method, with applications to understanding the stability of energy landscapes in disordered systems.

Contribution

It introduces a replica-based approach to compute the Hessian index distribution without assuming specific matrix ensemble properties, applicable to non-Gaussian cases.

Findings

Derived the distribution of negative eigenvalues for Gaussian matrices

Applicable to stability analysis in disordered energy landscapes

Method extendable to non-Gaussian matrix ensembles

Abstract

We compute the distribution of the number of negative eigenvalues (the index) for an ensemble of Gaussian random matrices, by means of the replica method. This calculation has important applications in the context of statistical mechanics of disordered systems, where the second derivative of the potential energy (the Hessian) is a random matrix whose negative eigenvalues measure the degree of instability of the energy surface. An analysis of the probability distribution of the Hessian index is therefore relevant for a geometric characterization of the energy landscape in disordered systems. The approach we use here is particularly suitable for this purpose, since it addresses the problem without any a priori assumption on the random matrix ensemble and can be naturally extended to more realistic, non-Gaussian distributions.