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Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition

Abstract

Consider the product GX of two rectangular complex random matrices coupled by a constant matrix \(\Omega \), where G can be thought to be a Gaussian matrix and X is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin’s sense, and further that for X being Gaussian, the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of \(\Omega ^{} \Omega ^{*}\) are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel, and the finite coupled product kernel associated with GX. In the special case when X is also a Gaussian matrix and \(\Omega \) is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.

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Acknowledgements

We are grateful to Gernot Akemann, Peter J. Forrester, Jiang Hu, Mario Kieburg, Dong Wang, and Lun Zhang for helpful discussions. We also thank the anonymous referees for their careful reading and constructive suggestions. The work was partially supported by the National Natural Science Foundation of China #11771417, the Youth Innovation Promotion Association CAS #2017491, Anhui Provincial Natural Science Foundation #1708085QA03 and the Fundamental Research Funds for the Central Universities (Grants WK0010450002 and WK3470000008).

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Correspondence to Dang-Zheng Liu.

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Communicated by Arno Kuijlaars.

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Liu, DZ. Singular Values for Products of Two Coupled Random Matrices: Hard Edge Phase Transition. Constr Approx 47, 487–528 (2018). https://doi.org/10.1007/s00365-017-9389-z

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  • DOI: https://doi.org/10.1007/s00365-017-9389-z

Keywords

  • Products of random matrices
  • Bi-orthogonal ensembles
  • Meijer G-kernel
  • Hard edge limit
  • Phase transition

Mathematics Subject Classification

  • 60B20
  • 62H25