Hard edge limit of the product of two strongly coupled random matrices

Abstract

We investigate the hard edge scaling limit of the ensemble defined by the\nsquared singular values of the product of two coupled complex random matrices.\nWhen taking the coupling parameter to be dependent on the size of the product\nmatrix, in a certain double scaling regime at the origin the two matrices\nbecome strongly coupled and we obtain a new hard edge limiting kernel. It\ninterpolates between the classical Bessel-kernel describing the hard edge\nscaling limit of the Laguerre ensemble of a single matrix, and the Meijer\nG-kernel of Kuijlaars and Zhang describing the hard edge scaling limit for the\nproduct of two independent Gaussian complex matrices. It differs from the\ninterpolating kernel of Borodin to which we compare as well.\n