The isotropic semicircle law and deformation of wigner matrices

Abstract

We analyze the spectrum of additive finite-rank deformations of N × N Wigner matrices H. The spectrum of the deformed matrix undergoes a transition, associated with the creation or annihilation of an outlier, when an eigenvalue di of the deformation crosses a critical value ± 1. This transition happens on the scale |di|-1∼N-1/3. We allow the eigenvalues di of the deformation to depend on N under the condition {norm of matrix}di| -1 |≥(logN)Clog log NN-1/3. We make no assumptions on the eigenvectors of the deformation. In the limit N → ∞, we identify the law of the outliers and prove that the nonoutliers close to the spectral edge have a universal distribution coinciding with that of the extremal eigenvalues of a Gaussian matrix ensemble. A key ingredient in our proof is the isotropic local semicircle law, which establishes optimal high-probability bounds on 〈 v,((H-z)-1-m(z)1)W 〉 where m(z) is the Stieltjes transform of Wigner's semicircle law and v, w are arbitrary deterministic vectors. © 2013 Wiley Periodicals, Inc.