The Isotropic Semicircle Law and Deformation of Wigner Matrices

Abstract

We analyse the spectrum of additive finite-rank deformations of $N\times N$ 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 $\pm 1$. This transition happens on the scale $|d_i|-1\sim N^{-1/3}$. We allow the eigenvalues $d_i$ of the deformation to depend on $N$ under the condition $|\abs{d_i} - 1| \geq (\log N)^{C \log \log N} N^{-1/3}$. We make no assumptions on the eigenvectors of the deformation. In the limit $N\to \infty$, we identify the law of the outliers and prove that the non-outliers 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 \emph{isotropic local semicircle law}, which establishes optimal high-probability bounds on the quantity $<v,[(H-z{)}^{-1}-m(z)1]w>$, where $m(z)$ is the Stieltjes transform of Wigner's semicircle law and $v,w$ are arbitrary deterministic vectors.

References in corpus (1)

• Local semicircle law and complete delocalization for Wigner random matrices

Cited by in corpus (10)

• Edge universality of correlation matrices
• Finding Hidden Cliques of Size \sqrt{N/e} in Nearly Linear Time
• Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices
• On the principal components of sample covariance matrices
• Low rank perturbations of large elliptic random matrices
• Spectra of nearly Hermitian random matrices
• Fluctuations at the edges of the spectrum of the full rank deformed GUE
• On Finite Rank Deformations of Wigner Matrices II: Delocalized Perturbations
• On a class of H-selfadjont random matrices with one eigenvalue of nonpositive type
• On the spectral properties of a class of $H$-selfadjoint random matrices and the underlying combinatorics