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Circular Jacobi ensembles and deformed Verblunsky coefficients


Bourgade, P; Nikeghbali, A; Rouault, A (2009). Circular Jacobi ensembles and deformed Verblunsky coefficients. International Mathematics Research Notices, 2009(23):4357-4394.

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: $$c_{\delta,\beta}^{(n)} \prod_{1\leq k<l\leq n}| e^{\ii\theta_k}-e^{\ii\theta_l}|^\beta\prod_{j=1}^{n}(1-e^{-\ii\theta_j})^{\delta} (1-e^{\ii\theta_j})^{\overline{\delta}} $$ with $\Re \delta > -1/2$. 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.

Formula
with Formula . If e is a cyclic vector for a unitary n x 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 us to prove that, in the regime {delta} = {delta} (n) with {delta} (n)/ n -> β d/2, 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.

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: $$c_{\delta,\beta}^{(n)} \prod_{1\leq k<l\leq n}| e^{\ii\theta_k}-e^{\ii\theta_l}|^\beta\prod_{j=1}^{n}(1-e^{-\ii\theta_j})^{\delta} (1-e^{\ii\theta_j})^{\overline{\delta}} $$ with $\Re \delta > -1/2$. 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.

Formula
with Formula . If e is a cyclic vector for a unitary n x 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 us to prove that, in the regime {delta} = {delta} (n) with {delta} (n)/ n -> β d/2, 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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2009
Deposited On:05 Jan 2010 13:55
Last Modified:05 Apr 2016 13:36
Publisher:Oxford University Press
ISSN:1073-7928
Publisher DOI:https://doi.org/10.1093/imrn/rnp092
Related URLs:http://arxiv.org/abs/0804.4512
Permanent URL: https://doi.org/10.5167/uzh-25037

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