Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices
Abstract
Consider a deterministic selfadjoint 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 nonperturbed 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 socalled matrix models.
 Publication:

arXiv eprints
 Pub Date:
 September 2010
 arXiv:
 arXiv:1009.0145
 Bibcode:
 2010arXiv1009.0145B
 Keywords:

 Mathematics  Probability
 EPrint:
 42 pages, Electron. J. Prob., Vol. 16 (2011), Paper no. 60, pages 16211662