Integrals over the circular ensembles relating to classical domains

Feng, Zhiming; Song, Ji Ping

doi:10.1088/1751-8113/42/32/325204

Cited by 13 publications

(19 citation statements)

References 20 publications

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“…[9]). An example of relevance to the present study is an identity of Fyodorov and Khorozhenko [14] (see also [8]), which reads…”

Section: Duality Identitiesmentioning

confidence: 94%

Matrix averages relating to Ginibre ensembles

Forrester¹,

Rains²

2009

J. Phys. A: Math. Theor.

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The theory of zonal polynomials is used to compute the average of a Schur polynomial of argument AX, where A is a fixed matrix and X is from the real Ginibre ensemble. This generalizes a recent result of Sommers and Khorozhenko [J. Phys. A 42 (2009), 222002], and furthermore allows analogous results to be obtained for the complex and real quaternion Ginibre ensembles. As applications, the positive integer moments of the general variance Ginibre ensembles are computed in terms of generalized hypergeometric functions, these are written in terms of averages over matrices of the same size as the moment to give duality formulas, and the averages of the power sums of the eigenvalues are expressed as finite sums of zonal polynomials.

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“…[9]). An example of relevance to the present study is an identity of Fyodorov and Khorozhenko [14] (see also [8]), which reads…”

Section: Duality Identitiesmentioning

confidence: 94%

Matrix averages relating to Ginibre ensembles

Forrester¹,

Rains²

2009

J. Phys. A: Math. Theor.

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“…The goal of this Section is to evaluate the averages (1.2) in terms of hypergeometric functions of matrix argument and prove Proposition 1.1. This requires the use of tools from the theory of symmetric functions and we now introduce the relevant notation, see also [24,29].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Characteristic polynomials of random truncations: moments, duality and asymptotics

Serebryakov¹,

Simm²,

Dubach³

2021

Preprint

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We study moments of characteristic polynomials of truncated Haar distributed matrices from the three classical compact groups O(N), U(N) and Sp(2N). For finite matrix size we calculate the moments in terms of hypergeometric functions of matrix argument and give explicit integral representations highlighting the duality between the moment and the matrix size as well as the duality between the orthogonal and symplectic cases. Asymptotic expansions in strong and weak non-unitarity regimes are obtained. Using the connection to matrix hypergeometric functions, we establish limit theorems for the log-modulus of the characteristic polynomial evaluated on the unit circle.

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“…3]). To make contact with (1) for general θ > 0, it is again the τ = 1 case of (3) which is relevant, but now augmented by the inclusion of an extra Schur polynomial factor (see (12) below for its definition). A generalisation of the PDFs (1) permitting negative values is…”

Section: Statement Of the Problem And Summary Of Resultsmentioning

confidence: 99%

Selberg integral theory and Muttalib–Borodin ensembles

Forrester

Ipsen

2018

Advances in Applied Mathematics

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We study Muttalib-Borodin ensembles -particular eigenvalue PDFs on the half-linewith classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We show how the theory of the Selberg integral, involving also Jack and Schur polynomials, naturally leads to a multi-parameter generalisation of these particular Muttalib-Borodin ensembles, and also to the explicit form of underlying biorthogonal polynomials of a single variable. A suitable generalisation of the original definition of the Muttalib-Borodin ensemble allows for negative eigenvalues. In the cases of generalised Gaussian, symmetric Jacobi and Cauchy weights, we show that the problem of computing the normalisations and the biorthogonal polynomials can be reduced down to Muttalib-Borodin ensembles with classical weights on the positive half-line.

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Integrals over the circular ensembles relating to classical domains

Cited by 13 publications

References 20 publications

Matrix averages relating to Ginibre ensembles

Matrix averages relating to Ginibre ensembles

Characteristic polynomials of random truncations: moments, duality and asymptotics

Selberg integral theory and Muttalib–Borodin ensembles

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