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Products of Rectangular Random Matrices: Singular Values and Progressive Scattering

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 bi-orthogonal polynomials. This generalises 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. The correlation functions are given by a determinantal point process, where the kernel can be expressed in terms of Meijer $G$-functions. We compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices. The location of the endpoints of support for the latter are analysed in detail for general $M$. Finally, we consider the so-called ergodic mutual information, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multi-fold scattering.