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Large deviations of the extreme eigenvalues of random deformations of matrices
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  • Published: 06 July 2011

Large deviations of the extreme eigenvalues of random deformations of matrices

  • F. Benaych-Georges1,
  • A. Guionnet2 &
  • M. Maida3 

Probability Theory and Related Fields volume 154, pages 703–751 (2012)Cite this article

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  • 30 Citations

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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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Author information

Authors and Affiliations

  1. UPMC Université Paris 6, LPMA, Case courier 188, 4 Place Jussieu, 75252, Paris Cedex 05, France

    F. Benaych-Georges

  2. UMPA, ENS Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    A. Guionnet

  3. Laboratoire de Mathématiques, Faculté des Sciences, Université Paris-Sud, Bâtiment 425, 91405, Orsay Cedex, France

    M. Maida

Authors
  1. F. Benaych-Georges
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  2. A. Guionnet
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  3. M. Maida
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Corresponding author

Correspondence to F. Benaych-Georges.

Additional information

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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  • Received: 26 October 2010

  • Revised: 17 June 2011

  • Published: 06 July 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0382-3

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Keywords

  • Random matrices
  • Large deviations

Mathematics Subject Classification (2000)

  • 15A52
  • 60F10
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