Circular law for non-Hermitian block band matrices with slowly growing bandwidth

We consider the empirical eigenvalue distribution for a class of non-Hermitian random block tridiagonal matrices $T$ with independent entries. The matrix has $n$ blocks on the diagonal and each block has size $\ell_n$, so the whole matrix has size $n\ell_n$. We assume that the nonzero entries are i.i.d. with mean 0, variance 1 and having sufficiently high moments. We prove that when the entries have a bounded density, then whenever $\lim_{n\to\infty}\ell_n=\infty$ and $\ell_n=O(\operatorname{Poly}(n))$, the normalized empirical spectral distribution of $T$ converges almost surely to the circular law. The growing bandwidth condition $\lim _{n\to\infty}\ell_n=\infty$ is the optimal condition of circular law with small bandwidth. This confirms the folklore conjecture that the circular law holds whenever the bandwidth increases with the dimension, while all existing results for the circular law are only proven in the delocalized regime $\ell_n\gg n$.