The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s (2N)1 2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble an...
LetA=(a ij ) be ann ×n matrix whose entries fori≧j are independent random variables anda ji =a ij . Suppose that everya ij is bounded and for everyi>j we haveEa ij =μ,D 2 a ij =σ2 andEa ii =v. We establish a large deviation principle for the largest eigenvalue of a rank one deformation of a matrix from the GUE or GOE. As a corollary, we get another proof of the phenomenon, well-known in learning theory and finance, that the largest eigenvalue separates from the bulk when the perturbation is large enough. A large part of the paper is devoted to an auxiliary result on the continuity of spherical integrals in the case when one of the matrix is of rank one, as studied in one of our previous works. 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. We consider random Hermitian matrices with independent Gaussian entries Mij,i≤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 Open image in new window and its rank. This rank is also allowed to increase with N in some restricted way. A recently published Letter by Kota and Potbhare (1977) obtains the averaged spectrum of a large symmetric random matrix each element of which has a finite mean: their results disagree with two recent calculations which predict that under certain circumstances a single isolated eigenvalue splits off from the continuous semicircular distribution of eigenvalues associated with the random part of the matrix. This letter offers a simple re-derivation of this result and corrects the error in the work of Kota and Potbhare.