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In this short note, we extend the celebrated results of Tao and Vu, andKrishnapur on the universality of empirical spectral distributions to a wideclass of inhomogeneous complex random matrices, by showing that a technical andhard-to-verify Fourier domination assumption may be replaced simply by anatural uniform anti-concentration assumption.  Along the way, we show that inhomogeneous complex random matrices, whoseexpected squared Hilbert-Schmidt norm is quadratic in the dimension, and whoseentries (after symmetrization) are uniformly anti-concentrated at $0$ andinfinity, typically have smallest singular value $\Omega(n^{-1/2})$. The rate$n^{-1/2}$ 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 acouple of anti-concentration inequalities for sums of independent, but notnecessarily identically distributed, complex random variables, which may alsobe useful in other contexts.

A note on the universality of ESDs of inhomogeneous random matrices