TY - JOUR

T1 - Circular jacobi ensembles and deformed verblunsky coefficients

AU - Bourgade, Paul

AU - Nikeghbali, Ashkan

AU - Rouault, Alain

PY - 2009

Y1 - 2009

N2 - 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 analog of the Jacobi ensemble: (Image Omitted) with ℛeδ > -1/2. If e is a cyclic vector for a unitary n × n matrix U, the spectral measure of the pair (U, e) is well parameterized by its Verblunsky coefficients (α0, ..., αn-1). We introduce here a deformation (γ0, ..., γn-1) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ0)⋯ r(γn-1) of elementary reflections parameterized by these coefficients. If γ0, ..., γn-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 δ = δ(n) with δ(n)/n → βd2, 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.

AB - 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 analog of the Jacobi ensemble: (Image Omitted) with ℛeδ > -1/2. If e is a cyclic vector for a unitary n × n matrix U, the spectral measure of the pair (U, e) is well parameterized by its Verblunsky coefficients (α0, ..., αn-1). We introduce here a deformation (γ0, ..., γn-1) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ0)⋯ r(γn-1) of elementary reflections parameterized by these coefficients. If γ0, ..., γn-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 δ = δ(n) with δ(n)/n → βd2, 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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U2 - 10.1093/imrn/rnp092

DO - 10.1093/imrn/rnp092

M3 - Article

AN - SCOPUS:77955876746

SN - 1073-7928

SP - 4357

EP - 4394

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 23

ER -