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Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition

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Abstract

The singular values squared of the random matrix product \({Y = {G_{r} G_{r-1}} \ldots G_{1} (G_{0} + A)}\), where each \({G_{j}}\) is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with the correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A*A are equal to bN, the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of \({0 < b < 1}\) is independent of b, and is in fact the same as that known for the case b =  0 due to Kuijlaars and Zhang. The critical regime of b =  1 allows for a double scaling limit by choosing \({{b = (1 - \tau/\sqrt{N})^{-1}}}\), and for this the critical kernel and outlier phenomenon are established. In the simplest case r =  0, which is closely related to non-intersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of \({b > 1}\) with two distinct scaling rates. Similar results also hold true for the random matrix product \({T_{r} T_{r-1} \ldots T_{1} (G_{0} + A)}\), with each \({T_{j}}\) being a truncated unitary matrix.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria, 3010, Australia

    Peter J. Forrester

  2. ARC Centre of Excellence for Mathematical and Statistical Frontiers, The University of Melbourne, Melbourne, Victoria, 3010, Australia

    Peter J. Forrester

  3. Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Dang-Zheng Liu

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  1. Peter J. Forrester
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  2. Dang-Zheng Liu
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Correspondence to Dang-Zheng Liu.

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Communicated by P. Deift

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Forrester, P.J., Liu, DZ. Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition. Commun. Math. Phys. 344, 333–368 (2016). https://doi.org/10.1007/s00220-015-2507-5

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