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2020

A note on the universality of ESDs of inhomogeneous random matrices

Jain, Vishesh, Silwal, Sandeep

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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.

A note on the universality of ESDs of inhomogeneous random matrices · Around

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Eigenvalues and condition numbers of random matrices
A. Edelman · 1988
Earlier work this paper cites.
Invertibility of random matrices: norm of the inverse
M. Rudelson · 2008
Earlier work this paper cites.
The Littlewood–Offord problem and invertibility of random matrices
M. Rudelson and R. Vershynin · 2008
Earlier work this paper cites.
Random matrices: the circular law
T. Tao and V. Vu · 2008
Earlier work this paper cites.

Similar

Smooth analysis of the condition number and the least singular value
T. Tao and V. Vu · 2010
Cited alongside, same era.
Random matrices: Universality of ESDs and the circular law
Original
T. Tao, V. Vu, and M. Krishnapur · 2010
Cited alongside, same era.
The smallest singular value of heavy-tailed not necessarily iid random matrices via random rounding
Original
G. V. Livshyts · 2018
Cited alongside, same era.

Then

Coverings of random ellipsoids, and invertibility of matrices with iid heavy-tailed entries
E. Rebrova and K. Tikhomirov · 2018
Later among the works it cites.
The strong circular law: a combinatorial view
V. Jain · 2019
Later among the works it cites.
The smallest singular value of inhomogeneous square random matrices
Original
G. V. Livshyts, K. Tikhomirov, and R. Vershynin · 2019
Later among the works it cites.

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