Circular Law for Random Block Band Matrices with Genuinely 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.

31 pages, 4 figures; minor corrections and updated references

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• Transition to chaos in random networks with cell-type-specific connectivity
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Cited by in corpus (1)

• CLT for non-Hermitian random band matrices with variance profiles