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

Abstract

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 variance bound or quantitative mixing assumption is required. A finite-moment comparison extends the result to bounded-density entries at explicit larger bandwidths, including nonperiodic graph supports and inhomogeneous weights. Second, for periodic scalar strips with a flat diagonal core, we prove a least-singular-value estimate without a density assumption and deduce the circular law for real subgaussian atoms, including Rademacher entries, at explicit sublinear power bandwidths. The latter argument combines quantitative Littlewood--Offord counting with a separator-tree elimination and does not require full-block off-diagonal couplings. Both parts use coarse singular-value counting to control logarithmic integrability.

39 pages. The prior version has a major gap in the proof of a coarse local circular law. We cannot fix this gap. Thus we replace it with the current version which proves weaker estimates for more general variance profiles. The 1/2+c result is replaced by the 2/3+c result here. Also added an independent result on periodic band matrices with Rademacher entry laws