# 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.

## 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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Physical Sciences > General Mathematics English 2009 05 Jan 2010 13:55 29 Jul 2020 20:18 Oxford University Press 1073-7928 Green https://doi.org/10.1093/imrn/rnp092 http://arxiv.org/abs/0804.4512