Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices

Abstract

In this paper, we investigate the fluctuations of a unit eigenvector associated to an outlier in the spectrum of a spiked N × N complex Deformed Wigner matrix M N. M N is defined as follows: M N = W N / √ N +A N where W N is an N ×N Hermitian Wigner matrix whose entries have a law µ satisfying a Poincare inequality and the matrix A N is a block diagonal matrix, with an eigenvalue θ of multiplicity one, generating an outlier in the spectrum of M N. We prove that the fluctuations of the norm of the projection of a unit eigenvector corresponding to the outlier of M N onto a unit eigenvector corresponding to θ are not universal. Indeed, we take away a fit approximation of its limit from this norm and prove the convergence to zero as N goes to ∞ of the Levy-Prohorov distance between this rescaled quantity and the convolution of µ and a centered Gaussian distribution (whose variance may depend depend upon N and may not converge).