Products of rectangular random matrices: Singular values and progressive scattering

Abstract

We discuss the product of M rectangular random matrices with independent Gaussian entries, which have several applications, including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint probability density function is obtained using the Harish-Chandra-Itzykson-Zuber integration formula. Explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two-matrix model and the method of biorthogonal polynomials. This generalizes the classical result for the so-called Wishart-Laguerre Gaussian unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for the product of square matrices...