Summary: In this short note, we extend the celebrated results of Tao and Vu, and Krishnapur on the universality of empirical spectral distributions to a wide class of inhomogeneous complex random matrices, by showing that a technical and hard-to-verify Fourier domination assumption may be replaced simply by a natural uniform anti-concentration assumption.
Along the way, we show that inhomogeneous complex random matrices, whose expected squared Hilbert-Schmidt norm is quadratic in the dimension, and whose entries (after symmetrization) are uniformly anti-concentrated at 0 and infinity, typically have smallest singular value \(\Omega(n^{-1/2})\). The raten \(n^{-1/2}\) is sharp, and closes a gap in the literature.
Our proofs closely follow recent works of Livshyts, and Livshyts, Tikhomirov, and Vershynin on inhomogeneous real random matrices. The new ingredient is an anti-concentration inequality for sums of independent, but not necessarily identically distributed, complex random variables, which may also be useful in other contexts.