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| HAL: hal-00169167, version 3 |
| arXiv: 0709.0036 |
| DOI: 10.1007/s10959-010-0285-8 |
| Detailed view |  Export this paper |
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| Journal of Theoretical Probability 23, 4 (2010) 945-950 |
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| Available versions: | v1 (2007-09-01) | v2 (2007-09-26) | v3 (2010-05-31) |
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| Circular law for non-central random matrices |
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| Djalil Chafai 1 |
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| (2010-11-06) |
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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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CNRS : UMR8050 – Université Paris XII - Paris Est Créteil Val-de-Marne – Université Paris XII - Paris Est Créteil Val-de-Marne |
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| Subject | : | Mathematics/Probability |
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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 | |
| From: Djalil Chafai | |
| Submitted on: Monday, 5 April 2010 12:26:01 | |
| Updated on: Saturday, 6 November 2010 00:45:08 | |
