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. <br /> 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.
Citation
Florent Benaych-Georges. Alice Guionnet. Mylène Maida. "Fluctuations of the Extreme Eigenvalues of Finite Rank Deformations of Random Matrices." Electron. J. Probab. 16 1621 - 1662, 2011. https://doi.org/10.1214/EJP.v16-929
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