Eigenvalue statistics for product complex Wishart matrices

Abstract

The eigenvalue statistics for complex N × N Wishart matrices Xr,s†, Xr,s, where Xr,sis equal to the product of r complex Gaussian matrices, and the inverse of s complex Gaussian matrices, are considered. In the case r = s the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function + s 1Fr . For finite N the eigenvalue probability density function is computed, and is shown to be an example of a biorthogonal ensemble. A double contour integral form of the corresponding correlation kernel is derived, which allows the hard edge scaled limit ..