The least singular value of R*XT + U*YV, after scaling by N, converges in distribution to 1 - e^{-r^2}, the same limit as for a complex Gaussian matrix.
Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble
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abstract
We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\in\mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where $z$ is around the spectral edge thus improving the classical bound of [Sankar, Spielman, Teng, 2006] in the edge regime. Lacking Br\'ezin-Hikami formulas in the real case, we rely on the superbosonization formula [Littelmann, Sommers, Zirnbauer, 2008].
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Universality of the least singular value for the sum of random matrices
The least singular value of R*XT + U*YV, after scaling by N, converges in distribution to 1 - e^{-r^2}, the same limit as for a complex Gaussian matrix.