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| HAL : hal-00310528, version 5 |
| arXiv : 0808.1502 |
| DOI : 10.1007/s00440-010-0336-1 |
| Fiche détaillée |  Récupérer au format |
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| Probability Theory and Related Fields (2011) OnlineFirst version to be published in a printed volume soon |
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| Versions disponibles : | v1 (11-08-2008) | v2 (11-08-2008) | v3 (11-08-2008) | v4 (11-08-2008) | v5 (09-06-2010) |
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| Circular Law Theorem for Random Markov Matrices |
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| Charles Bordenave 1Pietro Caputo 2 |
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| (11/01/2011) |
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| Consider an nxn random matrix X with i.i.d. nonnegative entries with bounded density, mean m, and finite positive variance sigma^2. Let M be the nxn random Markov matrix with i.i.d. rows obtained from X by dividing each row of X by its sum. In particular, when X11 follows an exponential law, then M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Our main result states that with probability one, the counting probability measure of the complex spectrum of n^(1/2)M converges weakly as n tends to infinity to the uniform law on the centered disk of radius sigma/m. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent. |
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| 1 : | Institut de Mathématiques de Toulouse (IMT) |
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Université Paul Sabatier - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées de Toulouse – CNRS : UMR5219 |
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| 2 : | Dipartimento di Matematica [Roma TRE] |
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Università degli Studi Roma TRE |
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| 3 : | Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA) |
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CNRS : UMR8050 – Université Paris XII - Paris Est Créteil Val-de-Marne – Université Paris XII - Paris Est Créteil Val-de-Marne |
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| Domaine | : | Mathématiques/Probabilités Mathématiques/Théorie spectrale |
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| Random matrices – Eigenvalues – Spectrum – Stochastic Matrices – Markov chains |
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| Liste des fichiers attachés à ce document : | ||||||||||
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| hal-00310528, version 5 | |
| http://hal.archives-ouvertes.fr/hal-00310528 | |
| oai:hal.archives-ouvertes.fr:hal-00310528 | |
| Contributeur : Djalil Chafai | |
| Soumis le : Mardi 8 Juin 2010, 21:05:53 | |
| Dernière modification le : Jeudi 10 Novembre 2011, 21:15:19 | |
