Large deviations of the extreme eigenvalues of random deformations of matrices

Abstract

Consider a real diagonal deterministic matrix Xn of size n with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale n, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of Xn converge to the edges of the support of the limiting measure and when we allow some eigenvalues of Xn, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when Xn is random, with law proportional to e-n Tr V(X) dX, for V growing fast enough at infinity and any perturbation of finite rank. © 2011 Springer-Verlag.