The largest eigenvalue of small rank perturbations of Hermitian random matrices

Abstract

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with independent Gaussian entries $M_{ij},i\le j$ with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large $N$ limit, various limiting distributions depending on both the eigenvalues of the matrix $(\mathbb{E}{M}_{ij}{)}_{i,j=1}^N$ and its rank.

Revised version with minor results added and shorter proofs. 44 pages, 4 figures