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

摘要

Based on the multivariate saddle point method we study the asymptotic behaviorof the characteristic polynomials associated to Wishart type random matrices thatare formed as products consisting of independent standard complex Gaussian and atruncated Haar distributed unitary random matrix. These polynomials form a generalclass of hypergeometric functions of type 2Fr. We describe the oscillatory behavioron the asymptotic interval of zeros by means of formulae of Plancherel–Rotach typeand subsequently use it to obtain the limiting distribution of the suitably rescaledzeros. Moreover, we show that the asymptotic zero distribution lies in the class ofRaney distributions and by introducing appropriate coordinates elementary and explicitcharacterizations are derived for the densities as well as for the distribution functions.