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

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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Correspondence to Dang-Zheng Liu.

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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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  • DOI: https://doi.org/10.1007/s00220-015-2507-5

Keywords

  • Random Matrice
  • Random Matrix
  • Random Matrix Theory
  • Eigenvalue Density
  • Global Density