The Zigzag Strategy for Random Band Matrices

TL;DR

This paper proves that broad classes of Hermitian random band matrices exhibit delocalized eigenfunctions and universal spectral statistics when the band width exceeds a critical threshold, using the zigzag strategy and multi-resolvent local laws.

Contribution

It introduces the zigzag strategy for analyzing random band matrices and extends results to general variance profiles and distributions, broadening previous work.

Findings

Eigenfunctions are fully delocalized in the bulk spectrum.

Eigenvalues follow Wigner-Dyson statistics.

Quantum unique ergodicity holds for general observables.

Abstract

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 $\sqrt{\eta}$-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.