Hua-type integrals over unitary groups and over projective limits of unitary groups

Neretin, Yury A.

doi:10.1215/s0012-7094-02-11423-9

Cited by 87 publications

(106 citation statements)

References 20 publications

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“…For β = 2, such measures were first considered by Hua [18] and Pickrell [28,29]. This case was also widely studied in [27] and [5] for its connections with the theory of representations and in [7] for its analogies with the Ewens measures on permutation groups.…”

Section: P Bourgade Et Almentioning

confidence: 99%

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Bourgade¹,

Nikeghbali²,

Rouault³

2009

International Mathematics Research Notices

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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: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 (α 0 , . . . , α n−1 ). We introduce here a deformation (γ 0 , . . . , γ n−1 ) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(γ 0 ) · · · r(γ n−1 ) of elementary reflections parameterized by these coefficients. If γ 0 , . . . , γ 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 δ = δ(n) with δ(n)/n → βd/2, the spectral measure and the empirical spectral distribution

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Section: P Bourgade Et Almentioning

confidence: 99%

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Bourgade¹,

Nikeghbali²,

Rouault³

2009

International Mathematics Research Notices

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“…Such samplings with δ ∈ R have already been studied on the finite-dimensional unitary group by Hua [18], and results about the infinite dimensional case (on complex Grassmannians) were given by Pickrell ([30] and [31]). More recently, Neretin [27] also considered this measure, introducing the possibility δ ∈ C. Borodin and Olshanski [7] have used the analogue of this measure in the framework of the infinite dimensional unitary group and proved ergodic properties. Forrester and Witte in [38] referred to this measure as the cJUE distribution.…”

Section: Questionmentioning

confidence: 99%

Ewens Measures on Compact Groups and Hypergeometric Kernels

Bourgade

Nikeghbali

Rouault

2010

Séminaire De Probabilités XLIII

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Summary. On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure, these factors become independent random variables with explicit distributions. Beyond the known results on the orthogonal and unitary groups (O(n) and U (n)), we treat the symplectic case. In U (n), this induces a family of probability changes analogous to the biassing in the Ewens sampling formula known for the symmetric group. Then we study the spectral properties of these measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The associated orthogonal polynomials give rise, as n tends to infinity to a limit kernel at the singularity.

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“…The space of virtual isometries we construct this way may be viewed as a natural extension of the space of virtual permutations of Kerov et al [5] as well as an extension of the space of virtual isometries of Neretin [9]. We then derive with purely probabilistic methods an almost sure convergence for these random matrices under the Haar measure: for a coherent Haar measure on virtual isometries, the smallest normalized eigenangles converge almost surely to a point process whose correlation function is given by the sine kernel.…”

mentioning

confidence: 99%

A Unitary Extension of Virtual Permutations

Bourgade¹,

Najnudel²,

Nikeghbali³

2012

International Mathematics Research Notices

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Analogously to the space of virtual permutations [5], we define projective limits of isometries: these sequences of unitary operators are natural in the sense that they minimize the rank norm between successive matrices of increasing sizes. The space of virtual isometries we construct this way may be viewed as a natural extension of the space of virtual permutations of Kerov, Olshanski and Vershik ([5]) as well as the space of virtual isometries of Neretin ([9]). We then derive with purely probabilistic methods an almost sure convergence for these random matrices under the Haar measure: for a coherent Haar measure on virtual isometries, the smallest normalized eigenangles converge a.s. to a point process whose correlation function is given by the sine kernel. This almost sure convergence actually holds for a larger class of measures as is proved by Borodin and Olshanski ([1]). We give a different proof, probabilist in the sense that it makes use of martingale arguments and shows how the eigenangles interlace when going from dimension n to n+ 1. Our method also proves that for some universal constant ε > 0, the rate of convergence is almost surely dominated by n −ε when the dimension n goes to infinity.

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Hua-type integrals over unitary groups and over projective limits of unitary groups

Cited by 87 publications

References 20 publications

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Ewens Measures on Compact Groups and Hypergeometric Kernels

A Unitary Extension of Virtual Permutations

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