Abstract
We study the asymptotic behavior of outliers in the spectrum of bounded rank perturbations of large random matrices. In particular, we consider perturbations of elliptic random matrices which generalize both Wigner random matrices and non-Hermitian random matrices with iid entries. As a consequence, we recover the results of Capitaine, Donati-Martin, and Feral for perturbed Wigner matrices as well as the results of Tao for perturbed random matrices with iid entries. Along the way, we prove a number of interesting results concerning elliptic random matrices whose entries have finite fourth moment; these results include a bound on the least singular value and the asymptotic behavior of the spectral radius.
Highlights
We investigate the asymptotic behavior of outliers in the spectrum of bounded rank perturbations of large random matrices
We begin by introducing the empirical spectral distribution of a square matrix
From Theorem 1.2 and Theorem 1.5, we see that the low rank perturbation AN does not effect the limiting empirical spectral distribution (ESD)
Summary
We investigate the asymptotic behavior of outliers in the spectrum of bounded rank perturbations of large random matrices. We begin by introducing the empirical spectral distribution of a square matrix. ΛN (M ) denote the eigenvalues of M. In this case, the empirical spectral measure μM is given by. If the matrix M is Hermitian, the eigenvalues λ1(M ), . In this case the ESD is given by. Given a random N × N matrix YN , an important problem in random matrix theory is to study the limiting distribution of the empirical spectral measure as N tends to infinity. We will use asymptotic notation, such as O, o, Ω, under the assumption that N → ∞.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
