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 ? ? (0, 1) so if bandwidth matrix X is at least n1?? and nonzero entries are iid variables mean zero slightly more than four finite moments, then limiting empirical eigenvalue distribution X, when properly normalized, converges in probability to uniform on unit disk complex plane. The key technical result singular value bound shifted bandwidth, which improves Cook [Ann. Probab. 46, 3442 (2018)] setting.