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Dropping the Independence: Singular Values for Products of Two Coupled Random Matrices

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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.

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Author information

Authors and Affiliations

  1. Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Gernot Akemann

  2. Université Paris-Sud, CNRS, LPTMS, UMR 8626, Bât. 100, 91405, Orsay, France

    Gernot Akemann

  3. Department of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Eugene Strahov

Authors
  1. Gernot Akemann
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  2. Eugene Strahov
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Correspondence to Gernot Akemann.

Additional information

Communicated by P. Deift

G. Akemann is supported partly by Investissements d’Avenir du LabEx PALM (ANR-10-LABX-0039-PALM) and by the SFB|TR12 “Symmetries and Universality in Mesoscopic Systems” of the German research council DFG. E. Strahov is supported in part by the Hebrew University Grant “Non-hermitian random matrices” No. 0337592.

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Akemann, G., Strahov, E. Dropping the Independence: Singular Values for Products of Two Coupled Random Matrices. Commun. Math. Phys. 345, 101–140 (2016). https://doi.org/10.1007/s00220-016-2653-4

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