Circular-law conjecture for non-Hermitian random band matrices

The conjecture asks when non-Hermitian random band matrices have eigenvalues distributed according to the circular law. The latest result concerns several specified variance profiles and entry distributions, rather than every admissible band matrix.

Known results

• $W \ge N^{1/2+c}$ for any $c>0$ under broad inhomogeneous band assumptions (August 2025).
• For block-tridiagonal matrices, $\ell_n \to \infty$ suffices under moment and density hypotheses (November 2025).
• For arbitrary doubly stochastic profiles, $\gamma>5/6$ in the Gaussian case and $\gamma>8/9$ in a subgaussian case (October 2024).
• Periodic block-band matrices were handled for $W \ge n^{32/33}\log n$ (August 2020).

September 2026 optimal-bandwidth result

A September 2026 report describes a paper proving the circular law at $W_N\to\infty$ for multiple periodic and continuous-profile models, and at $W_N\gg\log N$ for a discrete subgaussian model. This is a substantial advance toward the conjectured threshold, but the claim is unverified and does not cover the universal class.

Current status (as of September 2026): The optimal threshold is claimed for several specified model classes, while the universal conjecture for all non-Hermitian random band matrices remains open.