Dropping the independence: singular values for products of two coupled random matrices

Abstract

We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination of two independent, complex Gaussian random matrices, which is equivalent to a coupling of the two factors via an Itzykson-Zuber term. We prove that the squared singular values of such a product form a biorthogonal ensemble and establish its exact solvability. The parameter dependence allows us to interpolate between the singular value statistics of the Laguerre ensemble and that of the product of two independent complex Ginibre ensembles which are both known. We give exact formulae for the correlation kernel in terms of a complex double contour integral, suitable for the subsequent asymptotic analysis. In particular, we derive a Christoffel-Darboux type formula for the correlation kernel, based on a five term recurrence relation for our biorthogonal functions. It enables us to find its scaling limit at the origin representing a hard edge. The resulting limiting kernel coincides with the universal Meijer G-kernel found by several authors in different ensembles. We show that the central limit theorem holds for the linear statistics of the singular values and give the limiting variance explicitly.

38 pages, v2: 2 references added, v3: 1 typo corrected and grant acknowledgement added

References in corpus (7)

• Recent exact and asymptotic results for products of independent random matrices
• Differential equations for singular values of products of Ginibre random matrices
• Universality conjecture and results for a model of several coupled positive-definite matrices
• Singular values for products of complex Ginibre matrices with a source: hard edge limit and phase transition
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions
• Stressed Cooper pairing in QCD at high isospin density: effective Lagrangian and random matrix theory
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices

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• A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
• Hard edge limit of the product of two strongly coupled random matrices
• The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta =\frac{1}{2}$
• Matrix product ensembles of Hermite-type and the hyperbolic Harish-Chandra-Itzykson-Zuber integral
• Random Matrix Theory and Quantum Chromodynamics
• Spectral statistics for the difference of two Wishart matrices
• Singular values for products of two coupled random matrices: hard edge phase transition
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Averages of products and ratios of characteristic polynomials in polynomial ensembles
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• Large n limit for the product of two coupled random matrices
• Product matrix processes for coupled multi-matrix models and their hard edge scaling limits
• Integrable structure of products of finite complex Ginibre random matrices
• Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices