A localization-delocalization transition for nonhomogeneous random matrices

TL;DR

This paper studies how the eigenvectors of large sparse Gaussian random matrices transition from localized to delocalized states depending on the average number of nonzero entries per row, revealing a universal phase transition near the spectral edge.

Contribution

It establishes a universal localization-delocalization transition for sparse Gaussian matrices based on sparsity level, contrasting with the pattern-sensitive nature of exact eigenvectors.

Findings

Delocalized eigenvectors appear when d >> log N.

Localized eigenvectors occur when d << log N.

Transition point is near d ~ log N.

Abstract

We consider $N\times N$ self-adjoint Gaussian random matrices defined by an arbitrary deterministic sparsity pattern with $d$ nonzero entries per row. We show that such random matrices exhibit a canonical localization-delocalization transition near the edge of the spectrum: when $d\gg\log N$ the random matrix possesses a delocalized approximate top eigenvector, while when $d\ll\log N$ any approximate top eigenvector is localized. The key feature of this phenomenon is that it is universal with respect to the sparsity pattern, in contrast to the delocalization properties of exact eigenvectors which are sensitive to the specific sparsity pattern of the random matrix.