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September 2014 The outliers of a deformed Wigner matrix
Antti Knowles, Jun Yin
Ann. Probab. 42(5): 1980-2031 (September 2014). DOI: 10.1214/13-AOP855

Abstract

We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1–47; Ann. Inst. Henri Poincaré Probab. Stat. 48 (1013) 107–133; Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Preprint]. We also show that, under suitable conditions, outliers may be strongly correlated even if they are far from each other. Our proof relies on the isotropic local semicircle law established in [The isotropic semicircle law and deformation of Wigner matrices. Preprint]. The main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.

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Antti Knowles. Jun Yin. "The outliers of a deformed Wigner matrix." Ann. Probab. 42 (5) 1980 - 2031, September 2014. https://doi.org/10.1214/13-AOP855

Information

Published: September 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1306.15034
MathSciNet: MR3262497
Digital Object Identifier: 10.1214/13-AOP855

Subjects:
Primary: 15B52, 60B20, 82B44

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 5 • September 2014
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