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.
Edge Universality for non-Hermitian Random Matrices
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abstract
We consider large non-Hermitian real or complex random matrices $X$ with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of $X$ are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy-Widom distribution at the spectral edges of the Wigner ensemble.
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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.