Exponential bounds for the support convergence in the Single Ring Theorem

Abstract : We consider an $n$ by $n$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix. We prove that for $k\sim n^{1/6}$ and $b:=\frac{1}{n}\operatorname{Tr}(|T|^2)$, as $n$ tends to infinity, we have $$\mathbb{E} \operatorname{Tr} (A^{k}(A^{k})^*) \ \lesssim \ b^{2k}\qquad \textrm{ and } \qquad\mathbb{E}[|\operatorname{Tr} (A^{k})|^2] \ \lesssim \ b^{2k}.$$ This gives a simple proof of the convergence of the support in the Single Ring Theorem, improves the available error bound for this convergence from $n^{-\alpha}$ to $e^{-cn^{1/6}}$ and proves that the rate of this convergence is at most $n^{-1/6}\log n$.
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Article dans une revue
Journal of Functional Analysis, Elsevier, 2015, 268 (11), pp.doi:10.1016/j.jfa.2015.03.005
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https://hal.archives-ouvertes.fr/hal-01069221
Contributeur : Florent Benaych-Georges <>
Soumis le : lundi 29 septembre 2014 - 06:21:46
Dernière modification le : vendredi 10 février 2017 - 12:34:39

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  • HAL Id : hal-01069221, version 1
  • ARXIV : 1409.3864

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Florent Benaych-Georges. Exponential bounds for the support convergence in the Single Ring Theorem. Journal of Functional Analysis, Elsevier, 2015, 268 (11), pp.doi:10.1016/j.jfa.2015.03.005. <hal-01069221>

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