Selberg integral theory and Muttalib–Borodin ensembles

Forrester, Peter J.; Ipsen, Jesper R.

doi:10.1016/j.aam.2017.11.004

Cited by 13 publications

(14 citation statements)

References 35 publications

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“…Very few results exist on MB determinants for general values of θ. It was noticed in [21,36] that MB determinants associated to the classical Jacobi and Laguerre weights are Selberg integrals which can be evaluated explicitly, and the asymptotics of MB determinants without FH singularities have been studied in [7]. To the best of our knowledge, for θ = 1 no results are available in the literature on the large n asymptotics of MB determinants whose weight has FH singularities in the interior of its support.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

Charlier

2018

International Mathematics Research Notices

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We obtain large n asymptotics of n × n Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. We restrict our attention to the case where the associated equilibrium measure possesses either one soft edge and one hard edge (Laguerretype) or two hard edges (Jacobi-type). We also present some applications in the theory of random matrices. In particular, we can deduce from our results asymptotics for partition functions with singularities, central limit theorems, correlations of the characteristic polynomials, and gap probabilities for (piecewise constant) thinned Laguerre and Jacobi-type ensembles. Finally, we mention some links with the topics of rigidity and Gaussian multiplicative chaos.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

Charlier

2018

International Mathematics Research Notices

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“…Ref. [17] shows how the corresponding biorthogonal polynomials relate to the theory of the Selberg integrals, while a double integral form of the correlation kernel is given in [19].…”

Section: The Case Of Induced Real Jacobi Matricesmentioning

confidence: 99%

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Kieburg

Forrester

Ipsen

2019

Advances in Pure and Applied Mathematics

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AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

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“…Moreover, the M = 1 case is the so-called Laguerre Muttalib-Borodin ensemble for which the biorthogonal polynomials are known. In particularly, we know from [30,40,13] that…”

Section: Initial Matrix As An Elementary Anti-symmetric Matrix and Prmentioning

confidence: 99%

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

Forrester

Ipsen

Liu

et al. 2019

Random Matrices: Theory Appl.

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In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and particular Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure X M · · · X 1 (iA)X T 1 · · · X T M , where each X i is a standard real Gaussian matrix, and A is a real anti-symmetric matrix can be determined. For M = 1 and A the bidiagonal anti-symmetric matrix with 1's above the diagonal, this reclaims results of Defosseux. For general M , and this choice of A, or A itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

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Selberg integral theory and Muttalib–Borodin ensembles

Cited by 13 publications

References 35 publications

Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

Asymptotics of Hankel Determinants With a One-Cut Regular Potential and Fisher–Hartwig Singularities

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

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