The Isotropic Semicircle Law and Deformation of Wigner Matrices

2011/10/28 by Antti Knowles, Jun Yin, Knowles, Antti +1 · 12 citations
Mathematics · Physics and Astronomy · #15B52 #60B20 #82B44 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Combinatorics #Deformation (meteorology) #Distribution (mathematics) #Eigenvalues and eigenvectors #FOS: Mathematics #FOS: Physical sciences #Gaussian #Isotropy #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Physics #Probability (math.PR) #Quantum mechanics #Random Matrices and Applications #Random matrix #Rank (graph theory) #Spectrum (functional analysis) #math-ph #math.MP #math.PR #msc:15B52 #msc:60B20 #msc:82B44

paper · pdf · doi:10.48550/arxiv.1110.6449

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2011/10/28 · arxiv created 2012/05/22 · arxiv updated 2012/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 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 |\absd_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 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.

Citations

• Universality for generalized Wigner matrices with Bernoulli distribution
• Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices

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• On the spectral radius of the ratio of Girko matrices
• Convergence of local eigenvector processes of generalized Wigner matrices
• Optimal Delocalization for Non--Hermitian Eigenvectors
• Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices
• Spectra of nearly Hermitian random matrices
• Spectra of contractions of the Gaussian Orthogonal Tensor Ensemble
• Signal detection from spiked noise via asymmetrization
• On a class of H-selfadjont random matrices with one eigenvalue of\n nonpositive type

Related

• The outliers of a deformed Wigner matrix
• On the Local Semicircular Law for Wigner Ensembles
• On Finite Rank Deformations of Wigner Matrices
• Lectures on the local semicircle law for Wigner matrices
• The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations