Localization and top eigenvalue detection

TL;DR

This paper develops a new cavity method to detect the top eigenvalue in random matrices, especially when the eigenvector is localized, overcoming limitations of previous approaches.

Contribution

It introduces a novel criterion within the cavity method framework to identify the top eigenvalue for localized eigenvectors, validated on the Anderson model.

Findings

Effective detection of localized top eigenvectors

Validated approach on Anderson model

Overcomes previous method limitations

Abstract

The detection of the top eigenvalue and its corresponding eigenvector in ensembles of random matrices has significant applications across various fields. An existing method, based on the linear stability of a complementary set of cavity equations, has been successful in identifying the top eigenvalue when the associated eigenvector is extended. However, this approach fails when the eigenvector is localized. In this work, we adapt the real-valued cavity method to address this limitation by introducing a novel criterion that exploits the constraints of the cavity equations to detect the top eigenvalue in systems with a localized top eigenvector. Our results are validated using the Anderson model as a paradigmatic example.