The Isotropic Semicircle Law and Deformation of Wigner Matrices

We analyze the spectrum of additive finite-rank deformations of N x 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 d(i) of the deformation crosses a critical value +/- 1. This transition happens on the scale vertical bar d(j)vertical bar - 1 similar to N-1/3. We allow the eigenvalues d(i) of the deformation to depend on N under the condition vertical bar vertical bar d(i)vertical bar - 1 vertical bar >= (log N)(C) (log) (log) (N) N-1/3. We make no assumptions on the eigenvectors of the deformation. In the limit N -> infinity, 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. (C) 2013 Wiley Periodicals, Inc.