  The product of M complex random Gaussian matrices of size N has recently been
studied by Akemann, Kieburg and Wei. They showed that, for fixed M and N, the
joint probability distribution for the squared singular values of the product
matrix forms a determinantal point process with a correlation kernel determined
by certain biorthogonal polynomials that can be explicitly constructed. We find
that, in the case M=2, the relevant biorthogonal polynomials are actually
special cases of multiple orthogonal polynomials associated with Macdonald
functions (modified Bessel functions of the second kind) which was first
introduced by Van Assche and Yakubovich. With known results on asymptotic zero
distribution of these polynomials and general theory on multiple orthogonal
polynomial ensembles, it is then easy to obtain an explicit expression for the
distribution of squared singular values for the product of two complex random
Gaussian matrices in the limit of large matrix dimensions.
