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

Thorsten Neuschel and Dries Stivigny

Journal of Multivariate Analysis, 2016, vol. 147, issue C, 155-167

Abstract: 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 2Fr. 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.

Keywords: Asymptotics; Multivariate saddle point method; Asymptotic distribution of zeros; Macroscopic density of eigenvalues; Plancherel–Rotach formula; Raney distribution; Ginibre random matrices; Complex Gaussian matrices, truncated unitary matrices; Average characteristic polynomials; Generalized hypergeometric polynomials (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1016/j.jmva.2016.01.008

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