Jacobi polynomial moments and products of random matrices

Gawronski, Wolfgang; Neuschel, Thorsten; Stivigny, Dries

doi:10.1090/proc/13153

Cited by 7 publications

(14 citation statements)

References 40 publications

(42 reference statements)

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“…Let G j denote a standard complex Gaussian matrix of size (n + ν j ) × (n + ν j−1 ). According to the recent work [27], in the limit n → ∞ the singular values squared of the random matrix product G r · · · G s+1 T s · · · T 1 (3.43)…”

Section: )mentioning

confidence: 99%

Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Forrester¹,

Wang²

2017

Electron. J. Probab.

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Muttalib-Borodin ensembles are characterised by the pair interaction term in the eigenvalue probability density function being of the formWe study the Laguerre and Jacobi versions of this model -so named by the form of the one-body interaction termsand show that for θ ∈ Z + they can be realised as the eigenvalue PDF of certain random matrices with Gaussian entries. For general θ > 0, realisations in terms of the eigenvalue PDF of ensembles involving triangular matrices are given. In the Laguerre case this is a recent result due to Cheliotis, although our derivation is different. We make use of a generalisation of a double contour integral formula for the correlation functions contained in a paper by Adler, van Moerbeke and Wang to analyse the global density (which we also analyse by studying characteristic polynomials), and the hard edge scaled correlation functions. For the global density functional equations for the corresponding resolvents are obtained; solving this gives the moments in terms of Fuss-Catalan numbers (Laguerre case -a known result) and particular binomial coefficients (Jacobi case). For θ ∈ Z + the Laguerre and Jacobi cases are closely related to the squared singular values for products of θ standard Gaussian random matrices, and truncations of unitary matrices, respectively. At the hard edge the double contour integral formulas provide a double contour integral form of the scaled correlation kernel obtained by Borodin in terms of Wright's Bessel function.

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Section: )mentioning

confidence: 99%

Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Forrester¹,

Wang²

2017

Electron. J. Probab.

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“…Let F (z) denote the Stieltjes transform of the measure µ r,s . It can be derived using notions from free probability [13] that the function w(z) = zF (z) satisfies the algebraic equation…”

Section: )mentioning

confidence: 99%

“…where x * > 0 is the right endpoint of the support of µ r,r (as the measure is the free multiplicative convolution of compactly supported measures on the positive real axis, x * has to be finite). Using properties of the S-transform from free probability theory it can be derived (see, e.g., [13]) that the function w(z) = zF (z) satisfies the algebraic equation Up to a scaling in the argument, this is the equation for the Stieltjes transforms in the case s = 0, which coincides with the Fuss-Catalan case. It is known (see, e.g., [11,21]) that the boundary values of v on the branch cut (0, x * ) can be stated explicitly by This equation is of a similar type as (3.8), which enables us to find a functional relation between G and the Stieltjes transform F of µ r,r in terms of a rational transformation F (z) = (r + 1)2 r G 2 r+1 z 1 + (r − 1)2 r zG (2 r+1 z) .…”

Section: Density Of Singular Values Of Products With At Most One Compmentioning

confidence: 99%

“…A powerful machinery to characterize limiting distributions of eigenvalues of products of random matrices is the notion of free multiplicative convolution which was developed in free probability theory (see, e.g., [26,28]). Using this approach it can be shown [13] that as n → ∞ the singular values of the rescaled product…”

Section: Introductionmentioning

confidence: 99%

“…The moments of µ r,s have recently been studied [13] showing that they can be found explicitly in terms of Jacobi polynomials by µ r,s (0) = 1 and for k ≥ 1…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral densities of singular values of products of Gaussian and truncated unitary random matrices

Neuschel

2019

Random Matrices: Theory Appl.

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We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth.In the general case we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian we derive elementary expressions for the limiting densities using suitable parameterizations for the spectral variable. Moreover, in all cases we study the behavior of the densities at the boundary of the spectrum.

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Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices

Neuschel

Stivigny

2016

Journal of Multivariate Analysis

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Based on the multivariate saddle point method we study the asymptotic behavior of the characteristic polynomials associated to Wishart type random matrices that are formed as products consisting of independent standard complex Gaussian and a truncated Haar distributed unitary random matrix. These polynomials form a general class of hypergeometric functions of type 2 F r . We describe the oscillatory behavior on the asymptotic interval of zeros by means of formulae of Plancherel-Rotach type and subsequently use it to obtain the limiting distribution of the suitably rescaled zeros. Moreover, we show that the asymptotic zero distribution lies in the class of Raney distributions and by introducing appropriate coordinates elementary and explicit characterizations are derived for the densities as well as for the distribution functions.

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Jacobi polynomial moments and products of random matrices

Cited by 7 publications

References 40 publications

Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Muttalib–Borodin ensembles in random matrix theory — realisations and correlation functions

Spectral densities of singular values of products of Gaussian and truncated unitary random matrices

Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices

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