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

TL;DR

This paper studies the behavior of the largest eigenvalues of deformed Wigner matrices, showing conditions for their convergence outside the semicircle support and demonstrating nonuniversality of fluctuations in certain cases.

Contribution

It provides new results on the asymptotic behavior of extreme eigenvalues of deformed Wigner matrices, including conditions for universality and nonuniversality of fluctuations.

Findings

Eigenvalues exit the semicircle support when eigenvalues of A_N are far from zero.

Universal limits depend only on the variance of W_N entries.

Fluctuations are nonuniversal when A_N has a large simple eigenvalue.

Abstract

In this paper, we investigate the asymptotic spectrum of complex or real Deformed Wigner matrices $(M_N)_N$ defined by $M_N=W_N/\sqrt{N}+A_N$ where $W_N$ is an $N\times N$ Hermitian (resp., symmetric) Wigner matrix whose entries have a symmetric law satisfying a Poincar\'{e} inequality. The matrix $A_N$ 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 $A_N$ are sufficiently far from zero, the corresponding eigenvalues of $M_N$ 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 $W_N$. On the other hand, when $A_N$ 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 $W_N$.