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 asymptoticbehavior of the characteristic polynomials associated to Wishart type randommatrices that are formed as products consisting of independent standard complexGaussian and a truncated Haar distributed unitary random matrix. Thesepolynomials form a general class of hypergeometric functions of type \(_2F_r\). We describe the oscillatory behavior on the asymptotic interval of zerosby means of formulae of Plancherel-Rotach type and subsequently use it toobtain the limiting distribution of the suitably rescaled zeros. Moreover, weshow that the asymptotic zero distribution lies in the class of Raneydistributions and by introducing appropriate coordinates elementary andexplicit characterizations are derived for the densities as well as for thedistribution functions.