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2011 Fluctuations of the Extreme Eigenvalues of Finite Rank Deformations of Random Matrices
Florent Benaych-Georges, Alice Guionnet, Mylène Maida
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Electron. J. Probab. 16: 1621-1662 (2011). DOI: 10.1214/EJP.v16-929


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.


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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.


Accepted: 31 August 2011; Published: 2011
First available in Project Euclid: 1 June 2016

zbMATH: 1245.60007
MathSciNet: MR2835249
Digital Object Identifier: 10.1214/EJP.v16-929

Primary: 60B20
Secondary: 60F05

Keywords: extreme eigenvalue statistics , Gaussian fluctuations , random matrices , spiked models , Tracy-Widom laws

Vol.16 • 2011
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