Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth

TL;DR

This paper establishes the circular law for a class of non-Hermitian random block band matrices with sublinear bandwidth, showing eigenvalue distribution convergence under specific conditions.

Contribution

It proves the circular law for non-Hermitian band matrices with genuinely sublinear bandwidth, extending previous results with new singular value bounds.

Findings

Eigenvalue distribution converges to the uniform distribution on the unit disk.

Established a least singular value bound for shifted band matrices.

Extended circular law results to matrices with sublinear bandwidth.

Abstract

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $\tau \in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-\tau}$ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of $X$, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook in the band matrix setting.