Density of small singular values of the shifted real Ginibre ensemble

TL;DR

This paper provides an asymptotic formula for the density of small singular values of shifted real Ginibre matrices, revealing a transition from real to complex Ginibre behavior as the shift parameter varies.

Contribution

It derives a precise asymptotic density formula for small singular values of shifted real Ginibre matrices, extending known results to the real case and analyzing the transition with complex shifts.

Findings

The density formula matches that of complex Ginibre matrices away from the real axis.

Confirmed the transition from real to complex Ginibre ensembles for singular values.

Used superbosonization and saddle point analysis to derive the results.

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.