# Large deviations of the extreme eigenvalues of random deformations of matrices

Abstract : Consider a real diagonal deterministic matrix $X_n$ of size $n$ with spectral measure converging to a compactly supported probability measure. We perturb this matrix by adding a random finite rank matrix, with delocalized eigenvectors. We show that the joint law of the extreme eigenvalues of the perturbed model satisfies a large deviation principle in the scale $n$, with a good rate function given by a variational formula. We tackle both cases when the extreme eigenvalues of $X_n$ converge to the edges of the support of the limiting measure and when we allow some eigenvalues of $X_n$, that we call outliers, to converge out of the bulk. We can also generalise our results to the case when $X_n$ is random, with law proportional to $e^{- n Trace V(X)}\ud X,$ for $V$ growing fast enough at infinity and any perturbation of finite rank.
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Document type :
Journal articles
Probability Theory and Related Fields, Springer Verlag, 2012
Domain :

https://hal.archives-ouvertes.fr/hal-00505502
Contributor : Florent Benaych-Georges <>
Submitted on : Saturday, June 18, 2011 - 11:38:08 AM
Last modification on : Thursday, April 27, 2017 - 9:45:49 AM
Document(s) archivé(s) le : Monday, September 19, 2011 - 2:21:01 AM

### Files

PGDDrevised180611.pdf
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### Identifiers

• HAL Id : hal-00505502, version 4
• ARXIV : 1009.0135

### Citation

Florent Benaych-Georges, Alice Guionnet, Mylène Maïda. Large deviations of the extreme eigenvalues of random deformations of matrices. Probability Theory and Related Fields, Springer Verlag, 2012. 〈hal-00505502v4〉

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