The circular law for non-Hermitian random band matrices: optimal bandwidth, periodic profile and discrete law

We consider non-Hermitian random band matrices with growing bandwidth and study convergence of their empirical spectral distributions to the circular law. Let $N$ denote the matrix size and $W_N$ the bandwidth. From a universality perspective, it is conjectured that the circular law holds whenever $W_N\to \mathrm{\infty }$. Previous results have mainly required $W_N\gg N^{1\mathrm{/}2}$, with the principal exception of \cite{Han2511}, which proves the $W_N\to \mathrm{\infty }$ circular law for an open-boundary block-tridiagonal model. Here we prove the circular law at this optimal threshold for several periodic models and under the near-optimal condition $W_N\gg \log N$ for a genuinely discrete model. For the periodic hard-indicator profile and its uniform and polynomially tapered generalizations, we prove the circular law under bounded-density and finite-third-moment assumptions whenever $W_N\to \mathrm{\infty }$. At the same threshold, we prove the circular law for continuous, integrable profiles locally bounded below on every finite interval, including exponentially and Gaussian decaying profiles, with circular complex Gaussian entries. For the periodic full-block model, we prove the circular law under bounded-density and finite-third-moment assumptions whenever $W_N\to \mathrm{\infty }$. The finite-third-moment assumption in the bounded-density results can be weakened to a finite $(2+\alpha )$-moment assumption. For the same full-block model without a density assumption, we prove the circular law for centered variance-one real subgaussian atoms when $W_N\gg \log N$. The proof uses compositions of random transfer operators. An established high-band circular-law input on a short auxiliary ring calibrates the full exterior coefficient norm, and a boundary-uniform local comparison lifts this calibration to target rings even when $N\mathrm{/}{W}_N$ is arbitrarily large.