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

Abstract

We study eigenvectors in the deformed Gaussian unitary ensemble of random matrices , where is a random matrix from the 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.