The Outliers of a Deformed Wigner Matrix

Antti Knowles Harvard University Jun Yin University of Wisconsin

Probability mathscidoc:1608.28009

Annals of Probability, 42, (5), 1980-2031, 2014
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 Poincar\'{e} 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 $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.
Random matrix, universality, deformation, outliers
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  title={The Outliers of a Deformed Wigner Matrix},
  author={Antti Knowles, and Jun Yin},
  booktitle={Annals of Probability},
Antti Knowles, and Jun Yin. The Outliers of a Deformed Wigner Matrix. 2014. Vol. 42. In Annals of Probability. pp.1980-2031.
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