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arXiv · 2105.13720

Density of small singular values of the shifted real Ginibre ensemble

Giorgio Cipolloni·László Erdős·Dominik Schröder
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Abstract

We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter $z$ as the dimension tends to infinity. For $z$ away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in [arXiv:1908.01653]. On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter $z$ becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula [arXiv:0707.2929] in a regime where the main contribution comes from a three dimensional saddle manifold.

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Giorgio Cipolloni, László Erdős, Dominik Schröder. 2021-06-02. Density of small singular values of the shifted real Ginibre ensemble. https://doi.org/10.1007/s00023-022-01188-8

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