| Identifiant de l'article : |
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hal-00505497, version 5 |
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arXiv:1009.0145 |
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| Domaine : |
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Mathématiques/Probabilités
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| Titre : |
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Fluctuations of the extreme eigenvalues of finite rank deformations of random 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 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 delocalized 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. |
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Langue du texte
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Anglais |
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| Mots Clés : |
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Random matrices – largest eigenvalue – Limit theorems – Tracy-Widom law |
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| Classification : |
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15A52, 60F05 |
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| Commentaire : |
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42 pages, Electron. J. Prob., Vol. 16 (2011), Paper no. 60, pages 1621-1662. |
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