Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices

Abstract : Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X_n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X_n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.
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Pré-publication, Document de travail
42 pages, Electron. J. Prob., Vol. 16 (2011), Paper no. 60, pages 1621-1662. 2010
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Dernière modification le : vendredi 28 avril 2017 - 01:07:57
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  • HAL Id : hal-00505497, version 5
  • ARXIV : 1009.0145

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Florent Benaych-Georges, Alice Guionnet, Mylène Maïda. Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. 42 pages, Electron. J. Prob., Vol. 16 (2011), Paper no. 60, pages 1621-1662. 2010. 〈hal-00505497v5〉

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