Local inhomogeneous circular law

TL;DR

This paper establishes a local inhomogeneous circular law for large random matrices with independent, variably scaled entries, extending previous results to more general variance profiles and providing optimal local spectral density convergence.

Contribution

It proves the local inhomogeneous circular law for matrices with general variance profiles, advancing understanding of spectral distributions in non-uniform random matrices.

Findings

Spectral density converges locally on optimal scales.

Supports inhomogeneous limiting spectral distribution.

Extends circular law to matrices with non-identical variances.

Abstract

We consider large random matrices $X$ with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of $X$.