The largest eigenvalue of small rank perturbations of Hermitian random matrices

Details The largest eigenvalue of small rank perturbations of Hermitian random matrices The largest eigenvalue of small rank perturbations of Hermitian random matrices

2004-11-22

arxiv.org/abs/math/0411487v2

Mathematics

Probability

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

Scientific paper

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 $M_{ij}, i\leq j$ 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.

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