Circular Law For Random Block Band Matrices With Genuinely Sublinear Bandwidth

We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show that there exists tau is an element of (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 [Ann. Probab. 46, 3442 (2018)] in the band matrix setting.