preprint draft László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Volodymyr Riabov author 1949f904-edfb-11eb-afb5-e2dfddabb93b GradSch department LaEr department Random matrices beyond Wigner-Dyson-Mehta project We prove that a very general class of $N\times N$ Hermitian random band matrices is in the delocalized phase when the band width $W$ exceeds the critical threshold, $W\gg \sqrt{N}$. In this regime, we show that, in the bulk spectrum, the eigenfunctions are fully delocalized, the eigenvalues follow the universal Wigner-Dyson statistics, and quantum unique ergodicity holds for general diagonal observables with an optimal convergence rate. Our results are valid for general variance profiles, arbitrary single entry distributions, in both real-symmetric and complex-Hermitian symmetry classes. In particular, our work substantially generalizes the recent breakthrough result of Yau and Yin [arXiv:2501.01718], obtained for a specific complex Hermitian Gaussian block band matrix. The main technical input is the optimal multi-resolvent local laws -- both in the averaged and fully isotropic form. We also generalize the $\sqrtη$-rule from [arXiv:2012.13215] to exploit the additional effect of traceless observables. Our analysis is based on the zigzag strategy, complemented with a new global-scale estimate derived using the static version of the master inequalities, while the zig-step and the a priori estimates on the deterministic approximations are proven dynamically. 2025 eng 10.48550/ARXIV.2506.06441 https://research-explorer.ista.ac.at/record/20575 Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band Matrices.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/ARXIV.2506.06441">https://doi.org/10.48550/ARXIV.2506.06441</a>. Erdös, L., &#38; Riabov, V. (n.d.). The zigzag strategy for random band matrices. <i>arXiv</i>. <a href="https://doi.org/10.48550/ARXIV.2506.06441">https://doi.org/10.48550/ARXIV.2506.06441</a> Erdös, László, and Volodymyr Riabov. “The Zigzag Strategy for Random Band Matrices.” <i>ArXiv</i>, doi:<a href="https://doi.org/10.48550/ARXIV.2506.06441">10.48550/ARXIV.2506.06441</a>. Erdös L, Riabov V. The zigzag strategy for random band matrices. arXiv, <a href="https://doi.org/10.48550/ARXIV.2506.06441">10.48550/ARXIV.2506.06441</a>. L. Erdös, V. Riabov, ArXiv (n.d.). L. Erdös and V. Riabov, “The zigzag strategy for random band matrices,” <i>arXiv</i>. . Erdös L, Riabov V. The zigzag strategy for random band matrices. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/ARXIV.2506.06441">10.48550/ARXIV.2506.06441</a> 205762025-10-29T19:09:03Z2026-04-07T12:32:19Z