The outliers of a deformed Wigner matrix

TL;DR

This paper characterizes the joint distribution of outlier eigenvalues in deformed Wigner matrices, revealing non-universality and correlations among outliers, extending previous results to overlapping outliers.

Contribution

It provides the first joint asymptotic distribution of all outliers in deformed Wigner matrices, including overlapping ones, and explores their correlations.

Findings

Outliers can be strongly correlated even when far apart.

The joint distribution of outliers is explicitly derived.

Universality can fail in the presence of overlaps.

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 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 $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.