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 squared singular values of the product of two coupled complex random matrices. When taking the coupling parameter to be dependent on the size of the product matrix, in a certain double scaling regime at the origin the two matrices become strongly coupled and we obtain a new hard edge limiting kernel. It interpolates between the classical Bessel-kernel describing the hard edge scaling limit of the Laguerre ensemble of a single matrix, and the Meijer G-kernel of Kuijlaars and Zhang describing the hard edge scaling limit for the product of two independent Gaussian complex matrices. It differs from the interpolating kernel of Borodin to which we compare as well.

37 pages, 3 figures; v2: Theorem 1.6 sharpened, version to appear in Nonlinearity; v3: typo in Theorem 1.6 (c) corrected

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• Spectral statistics for the difference of two Wishart matrices
• Singular values for products of two coupled random matrices: hard edge phase transition
• Hurwitz numbers and matrix integrals labeled with chord diagrams
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Matrix integrals and Hurwitz numbers
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
• Large n limit for the product of two coupled random matrices
• Integrable structure of products of finite complex Ginibre random matrices
• Integrals of tau functions
• Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices