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| HAL : hal-00169167, version 3 |
| arXiv : 0709.0036 |
| DOI : 10.1007/s10959-010-0285-8 |
| Fiche détaillée |  Récupérer au format |
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| Journal of Theoretical Probability 23, 4 (2010) 945-950 |
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| Versions disponibles : | v1 (01-09-2007) | v2 (26-09-2007) | v3 (31-05-2010) |
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| Circular law for non-central random matrices |
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| Djalil Chafai 1 |
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| (06/11/2010) |
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| Let $(X_{jk})_{j,k\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean $0$ and variance $1$. Let $\la_{n,1},\ldots,\la_{n,n}$ be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution $\frac{1}{n}(\de_{\la_{n,1}}+\cdots+\de_{\la_{n,n}})$ converges weakly as $n\to\infty$ to the uniform law over the unit disc $\{z\in\dC;|z|\leq1\}$. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\leq j,k\leq n}$ provided that $\mathrm{Tr}(MM^*)=O(n^2)$ and $\mathrm{rank}(M)=O(n^\al)$ with $\al<1$. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems. |
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| 1 : | Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA) |
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Université Paris Est Marne-la-Vallée – Université Paris XII - Paris Est Créteil Val-de-Marne – CNRS : UMR8050 – Fédération de Recherche Bézout |
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| Domaine | : | Mathématiques/Probabilités |
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| Random matrices – Circular law |
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| hal-00169167, version 3 | |
| http://hal.archives-ouvertes.fr/hal-00169167 | |
| oai:hal.archives-ouvertes.fr:hal-00169167 | |
| Contributeur : Djalil Chafai | |
| Soumis le : Lundi 5 Avril 2010, 12:26:01 | |
| Dernière modification le : Samedi 6 Novembre 2010, 00:45:08 | |
