Density of small singular values of the shifted real Ginibre ensemble

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.