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Article Dans Une Revue Electronic Journal of Probability Année : 2011

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

Résumé

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 delocalised 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.

Dates et versions

hal-00659338 , version 1 (12-01-2012)

Identifiants

Citer

F. Benaych-Georges, A. Guionnet, M. Maïda. Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Electronic Journal of Probability, 2011, 16 (60), pp.1621-1662. ⟨10.1214/EJP.v16-929⟩. ⟨hal-00659338⟩
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