  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.
