2014
DOI: 10.1007/s00220-014-2064-3
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Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

Abstract: Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral … Show more

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Cited by 126 publications
(280 citation statements)
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References 40 publications
(78 reference statements)
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“…The polynomial (42) is a hypergeometric function and, thus, a Meijer G-function [14]. It agrees for certain values of the parameters L WL , L CL , and L J with known results [11,13]. What is completely new are the results for β = 1, 4 and k = 1 which are essentially the same polynomials.…”
Section: Application To Product Matricessupporting
confidence: 57%
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“…The polynomial (42) is a hypergeometric function and, thus, a Meijer G-function [14]. It agrees for certain values of the parameters L WL , L CL , and L J with known results [11,13]. What is completely new are the results for β = 1, 4 and k = 1 which are essentially the same polynomials.…”
Section: Application To Product Matricessupporting
confidence: 57%
“…These co-sets are also the fermionic part of the supermatrices involved in the superbosonization formula [22]. Since we only discuss the average of products of determinants and not ratios superbosonization reduces to bosonization only involving the circular ensembles (11). Let us recall the properties of a matrix U ∈ CβE(γk).…”
Section: Preliminariesmentioning
confidence: 99%
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“…Moreover, Nagao and Wadati [44] deduced the Bessel kernel by scaling the Jacobi ensemble at the hard edges, ±1 . Kuijlaars and Zhang [38] obtain a limiting kernel as a generalization of Bessel kernel by scaling the correlation kernel of complex Ginibre random matrices at the hard edge, see the references therein for more information. The theory of integrable kernels was put forward in [33].…”
Section: An Equivalent Expression Readsmentioning
confidence: 99%