Statistics of eigenvectors in the deformed Gaussian unitary ensemble of random matrices

TL;DR

This paper analytically investigates how the eigenvectors of deformed Gaussian unitary matrices behave, showing that different deterministic deformations can cause eigenvectors to transition from extended to localized states, supported by numerical validation.

Contribution

It provides the first analytical calculation of eigenvector moments and distributions in the deformed GUE, revealing how deterministic deformations influence eigenvector localization.

Findings

Eigenvector components' moments and distributions are analytically derived.

Specific deformations W can induce transitions from extended to localized eigenvectors.

Numerical simulations confirm the analytical predictions.

Abstract

We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices $H=W\tilde{H}W$, where $\tilde{H}$ is a random matrix from Gaussian unitary ensemble and $W$ is a deterministic diagonal matrix with positive entries. Using the supersymmetry approach we calculate analytically the moments and the distribution function of the eigenvectors components for a generic matrix $W$. We show that specific choices of $W$ can modify significantly the nature of the eigenvectors changing them from extended to critical to localized. Our analytical results are supported by numerical simulations.