The Annals of Probability

The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations

Mireille Capitaine, Catherine Donati-Martin, and Delphine Féral

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Abstract

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices (MN)N defined by $M_{N}=W_{N}/\sqrt{N}+A_{N}$ where WN is an N×N Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincaré inequality. The matrix AN is Hermitian (resp., symmetric) and deterministic with all but finitely many eigenvalues equal to zero. We first show that, as soon as the first largest or last smallest eigenvalues of AN are sufficiently far from zero, the corresponding eigenvalues of MN almost surely exit the limiting semicircle compact support as the size N becomes large. The corresponding limits are universal in the sense that they only involve the variance of the entries of WN. On the other hand, when AN is diagonal with a sole simple nonnull eigenvalue large enough, we prove that the fluctuations of the largest eigenvalue are not universal and vary with the particular distribution of the entries of WN.

Article information

Source
Ann. Probab. Volume 37, Number 1 (2009), 1-47.

Dates
First available in Project Euclid: 17 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1234881683

Digital Object Identifier
doi:10.1214/08-AOP394

Mathematical Reviews number (MathSciNet)
MR2489158

Zentralblatt MATH identifier
1163.15026

Subjects
Primary: 15A52 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems 60F05: Central limit and other weak theorems

Keywords
Deformed Wigner matrices asymptotic spectrum Stieltjes transform largest eigenvalues fluctuations nonuniversality

Citation

Capitaine, Mireille; Donati-Martin, Catherine; Féral, Delphine. The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations. Ann. Probab. 37 (2009), no. 1, 1--47. doi:10.1214/08-AOP394. https://projecteuclid.org/euclid.aop/1234881683


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