For any fixed n×n complex matrix of norm up to 2^{n^0.001}, the least singular value of the matrix plus i.i.d. centered unit-variance complex noise is smaller than η with probability at most C(ξ) α, for α as small as 2^{-n^0.001}.
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Quantitative invertibility of random matrices: a combinatorial perspective
For any fixed n×n complex matrix of norm up to 2^{n^0.001}, the least singular value of the matrix plus i.i.d. centered unit-variance complex noise is smaller than η with probability at most C(ξ) α, for α as small as 2^{-n^0.001}.