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 . We allow the eigenvalues di of the deformation to depend on N under the condition . 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 where m(z) is the Stieltjes transform of Wigner's semicircle law and v, w are arbitrary deterministic vectors.© 2013 Wiley Periodicals, Inc.