Large deviations of the extreme eigenvalues of random deformations of matrices |
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Authors: | F. Benaych-Georges A. Guionnet M. Maida |
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Affiliation: | 1. UPMC Universit?? Paris 6, LPMA, Case courier 188, 4 Place Jussieu, 75252, Paris Cedex 05, France 2. UMPA, ENS Lyon, 46 all??e d??Italie, 69364, Lyon Cedex 07, France 3. Laboratoire de Math??matiques, Facult?? des Sciences, Universit?? Paris-Sud, Batiment 425, 91405, Orsay Cedex, France
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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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