2011/10/28 by Antti Knowles, Jun Yin, Knowles, Antti +1 · 5 citations
Mathematics · #15B52 #60B20 #82B44 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1110.6449
openalex publication_date 2011/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We analyse 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 |\absdi - 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 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 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.