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| HAL: hal-00175643, version 3 |
| arXiv: 0709.4678 |
| DOI: 10.1016/j.jmva.2009.10.013 |
| Detailed view |  Export this paper |
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| Journal of Multivariate Analysis 101 (2010) 555-567 |
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| Available versions: | v1 (2007-09-28) | v2 (2007-10-17) | v3 (2009-11-02) |
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| The Dirichlet Markov Ensemble |
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| Djalil Chafai 1 |
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| (2010-01-01) |
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| We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $\mathbb{R}^{n^2}$. This probability space provides random Markov matrices, with i.i.d.\ rows following the Dirichlet distribution of mean $(1/n,\ldots,1/n)$. We show that if $\bM$ is such a random matrix, then the empirical distribution built from the singular values of$\sqrt{n}\,\bM$ tends as $n\to\infty$ to a Wigner quarter--circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of $\sqrt{n}\,\bM$ tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of $\bM$ is of order $1-1/\sqrt{n}$ when $n$ is large. |
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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 Mathematics/Spectral Theory |
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| Random matrices – Markov matrices – Dirichlet distributions – Spectral gap – singular values |
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| hal-00175643, version 3 | |
| http://hal.archives-ouvertes.fr/hal-00175643 | |
| oai:hal.archives-ouvertes.fr:hal-00175643 | |
| From: Djalil Chafai | |
| Submitted on: Monday, 2 November 2009 11:10:08 | |
| Updated on: Tuesday, 29 December 2009 10:46:46 | |
