Abstract
Consider a real diagonal deterministic matrix X n 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 X n converge to the edges of the support of the limiting measure and when we allow some eigenvalues of X n , that we call outliers, to converge out of the bulk. We can also generalise our results to the case when X n 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.
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This work was supported by the Agence Nationale de la Recherche grant ANR-08-BLAN-0311-03.
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Benaych-Georges, F., Guionnet, A. & Maida, M. Large deviations of the extreme eigenvalues of random deformations of matrices. Probab. Theory Relat. Fields 154, 703–751 (2012). https://doi.org/10.1007/s00440-011-0382-3
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DOI: https://doi.org/10.1007/s00440-011-0382-3