## Annals of Probability

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

#### 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

https://projecteuclid.org/euclid.aop/1234881683

Digital Object Identifier
doi:10.1214/08-AOP394

Mathematical Reviews number (MathSciNet)
MR2489158

Zentralblatt MATH identifier
1163.15026

#### 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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