G. Akemann, {J. R. Ipsen} andM. Kieburg [“Products of rectangular random matrices: singular values and progressive scattering”, Phys. Rev. E 88, Article ID 052118, 13 p. (2013)] showed recently that the squared singular values of a product of \(M\) complex Ginibre matrices are distributed according to a determinantal point process. The authors introduce the notion of a polynomial ensemble and show how their result can be interpreted as a transformation of polynomial ensembles. They also show that the squared singular values of the product of \(M-1\) complex Ginibre matrices with one truncated unitary matrix is a polynomial ensemble, and they derive a double integral representation for the correlation kernel associated with this ensemble. They use this to calculate the scaling limit at the hard edge, which turns out to be the same scaling limit as the one found byA. B. J. Kuijlaars andL. Zhang [Commun. Math. Phys. 332, No. 2, 759–781 (2014;Zbl 1303.15046)] for the squared singular values of a product of \(M\) complex Ginibre matrices. The final result is that these limiting kernels also appear as scaling limits for the biorthogonal ensembles ofA. Borodin [Nucl. Phys., B 536, No. 3, 704–732 (1999;Zbl 0948.82018)] with parameter \(\theta >0\), in case \(\theta\) or \(1/\theta\) is an integer. This further supports the conjecture that these kernels have a universal character.

Reviewer: A. Arvanitoyeorgos (Patras)

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