Circular Jacobi Ensembles and deformed Verblunsky coefficients

Abstract : Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)} \prod_{1\leq k -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $(\alpha_0, ..., \alpha_{n-1})$. We introduce here a deformation $(\gamma_0, >..., \gamma_{n-1})$ of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product $r(\gamma_0)... r(\gamma_{n-1})$ of elementary reflections parameterized by these coefficients. If $\gamma_0, ..., \gamma_{n-1}$ are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky coefficients also allow to prove that, in the regime $\delta = \delta(n)$ with $\delta(n)/n \to \dd$, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution.
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Article dans une revue
International Mathematics Research Notices, Oxford University Press (OUP), 2009, pp.4357-4394
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https://hal.archives-ouvertes.fr/hal-00672881
Contributeur : Nadège Arnaud <>
Soumis le : mercredi 22 février 2012 - 11:20:05
Dernière modification le : jeudi 27 avril 2017 - 09:47:30

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  • HAL Id : hal-00672881, version 1
  • ARXIV : 0804.4512

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Paul Bourgade, Ashkan Nikeghbali, Alain Rouault. Circular Jacobi Ensembles and deformed Verblunsky coefficients. International Mathematics Research Notices, Oxford University Press (OUP), 2009, pp.4357-4394. 〈hal-00672881〉

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