| Identifiant de l'article : |
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hal-00505502, version 4 |
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arXiv:1009.0135 |
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| Domaine : |
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Mathématiques/Probabilités
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| Titre : |
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Large deviations of the extreme eigenvalues of random deformations of matrices |
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| Auteur(s) : |
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Florent Benaych-Georges1, 2, Alice Guionnet3, Mylène Maïda4 |
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| Laboratoire : |
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LPMA - Laboratoire de Probabilités et Modèles Aléatoires |
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CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique |
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UMPA-ENSL - Unité de Mathématiques Pures et Appliquées |
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LM-Orsay - Laboratoire de Mathématiques d'Orsay |
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| Résumé : |
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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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Langue du texte
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Anglais |
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| Mots Clés : |
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Random matrices – Large deviations |
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| Classification : |
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15A52, 60F10 |
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| Commentaire : |
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44 pages |
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