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.
Document type :
Preprints, Working Papers, ...
42 pages, Electron. J. Prob., Vol. 16 (2011), Paper no. 60, pages 1621-1662. 2010
Liste complète des métadonnées

Cited literature [39 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00505497
Contributor : Florent Benaych-Georges <>
Submitted on : Friday, September 2, 2011 - 9:17:09 AM
Last modification on : Friday, April 28, 2017 - 1:07:57 AM
Document(s) archivé(s) le : Tuesday, November 13, 2012 - 9:45:29 AM

Files

fluctu_VPmax_ejp.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00505497, version 5
  • ARXIV : 1009.0145

Collections

Citation

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〉

Share

Metrics

Record views

341

Files downloads

56