A note on the universality of ESDs of inhomogeneous random matrices

TL;DR

This paper extends universality results for empirical spectral distributions of inhomogeneous complex random matrices by replacing a Fourier domination assumption with a natural anti-concentration condition, and establishes sharp bounds on the smallest singular value.

Contribution

It introduces a simplified anti-concentration condition for universality and provides sharp bounds on the smallest singular value of inhomogeneous complex matrices.

Findings

Universality of ESDs holds under anti-concentration assumptions.

Smallest singular value is typically of order n^{-1/2}.

New anti-concentration inequalities for sums of complex variables.

Abstract

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 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 a couple of anti-concentration inequalities for sums of independent, but not necessarily identically distributed, complex random variables, which may also be useful in other contexts.