@article{EJP929,
author = {Florent Benaych-Georges and Alice Guionnet and Mylène Maida},
title = {Fluctuations of the Extreme Eigenvalues of Finite Rank Deformations of Random Matrices},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {16},
year = {2011},
keywords = {random matrices ; spiked models ; extreme eigenvalue statistics ; Gaussian fluctuations ; Tracy-Widom laws},
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 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.},
pages = {no. 60, 1621-1662},
issn = {1083-6489},
doi = {10.1214/EJP.v16-929},
url = {http://ejp.ejpecp.org/article/view/929}}