The circular law for non-Hermitian random band matrices up to bandwidth $N^{2/3+c}$

We prove circular laws for non-Hermitian random matrices in two regimes. First, for doubly stochastic variance profiles bounded by $C/W$ and satisfying symmetry or transitive invariance, we establish the circular law for real or circular complex Gaussian entries when $W\ge N^{2/3+c}$. No lower varia