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The Isotropic Semicircle Law and Deformation of Wigner Matrices
Author(s) -
Knowles Antti,
Yin Jun
Publication year - 2013
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21450
Subject(s) - mathematics , eigenvalues and eigenvectors , isotropy , gaussian , matrix (chemical analysis) , deformation (meteorology) , spectrum (functional analysis) , random matrix , rank (graph theory) , mathematical analysis , law , combinatorics , quantum mechanics , physics , materials science , meteorology , political science , composite material
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 d i of the deformation crosses a critical value ± 1. This transition happens on the scale| d i | − 1 ∼ N − 1 / 3. We allow the eigenvalues d i of the deformation to depend on N under the condition‖ d i | − 1 |≥( log N )C log log N N − 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.