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

TL;DR

This paper analyzes the asymptotic behavior of characteristic polynomials of Wishart type matrices formed from complex Gaussian and truncated unitary matrices, revealing their zero distribution and connection to Raney distributions.

Contribution

It introduces a multivariate saddle point method to study these polynomials, describing their oscillatory behavior and limiting zero distribution in explicit terms.

Findings

Zeros follow Raney distributions

Explicit formulas for densities and distribution functions

Asymptotic oscillatory behavior characterized by Plancherel-Rotach type formulas

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 \(_2 F_r\). 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.