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We investigate the hard edge scaling limit of the ensemble defined by thesquared singular values of the product of two coupled complex random matrices.When taking the coupling parameter to be dependent on the size of the productmatrix, in a certain double scaling regime at the origin the two matricesbecome strongly coupled and we obtain a new hard edge limiting kernel. Itinterpolates between the classical Bessel-kernel describing the hard edgescaling limit of the Laguerre ensemble of a single matrix, and the MeijerG-kernel of Kuijlaars and Zhang describing the hard edge scaling limit for theproduct of two independent Gaussian complex matrices. It differs from theinterpolating kernel of Borodin to which we compare as well.

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