  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].
