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January 2009 The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations
Mireille Capitaine, Catherine Donati-Martin, Delphine Féral
Ann. Probab. 37(1): 1-47 (January 2009). DOI: 10.1214/08-AOP394

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.

Citation

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Mireille Capitaine. Catherine Donati-Martin. Delphine Féral. "The largest eigenvalues of finite rank deformation of large Wigner matrices: Convergence and nonuniversality of the fluctuations." Ann. Probab. 37 (1) 1 - 47, January 2009. https://doi.org/10.1214/08-AOP394

Information

Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1163.15026
MathSciNet: MR2489158
Digital Object Identifier: 10.1214/08-AOP394

Subjects:
Primary: 15A18‎, 15A52, 60F05, 60F15

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.37 • No. 1 • January 2009
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