Low rank perturbations of large elliptic random matrices. (English) Zbl 1315.60008

O’Rourke, Sean; Renfrew, David

Electron. J. Probab. 19, Paper No. 43, 65 p. (2014).

Summary: 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 M. Capitaine et al. [Ann. Probab. 37, No. 1, 1–47 (2009; Zbl 1163.15026); Ann. Inst. Henri Poincaré, Probab. Stat. 48, No. 1, 107–133 (2012; Zbl 1237.60007)] for perturbed Wigner matrices as well as the results of T. Tao [Probab. Theory Relat. Fields 155, No.  1–2, 231–263 (2013; Zbl 1261.60009); erratum ibid. 157, No.  1–2, 511–514 (2013; Zbl 1277.60018)] 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.

MSC:

60B20

Random matrices (probabilistic aspects)

Keywords:

elliptic random matrix; low rank perturbation; Wigner matrix

Citations:

Zbl 1163.15026; Zbl 1237.60007; Zbl 1261.60009; Zbl 1277.60018