Delocalization of random band matrices at the edge

TL;DR

This paper investigates how eigenvectors of Hermitian random band matrices in one and two dimensions transition from localized to delocalized states across the spectrum, extending known results to spectral edges and establishing quantum ergodicity.

Contribution

It extends previous bulk regime results to the entire spectrum, including edges, and establishes delocalization thresholds, quantum ergodicity, and eigenvalue rigidity for 1D and 2D random band matrices.

Findings

Eigenvectors near spectral edges become delocalized as band width increases.

Delocalization occurs for energies satisfying 2 - |E|  N^{-c_{d,\u03b1}} with specific exponents.

All eigenvectors are delocalized when  > 1 - d/6.

Abstract

We consider $N\times N$ Hermitian random band matrices $H=(H_{xy})$, whose entries are centered complex Gaussian random variables. The indices $x,y$ range over the $d$-dimensional discrete torus $(\mathbb Z/L\mathbb Z)^d$ with $d\in \{1,2\}$ and $N=L^d$. The variance profile $S_{xy}=\mathbb E|h_{xy}|^2$ exhibits a banded structure: specifically, $S_{xy}=0$ whenever the distance $|x-y|$ exceeds a band width parameter $W\le L$. Let $W=L^\alpha$ for some exponent $0<\alpha\le 1$. We show that as $\alpha$ increases from $\mathbf 1_{d=1}/2$ to $1-d/6$, the range of energies corresponding to delocalized eigenvectors gradually expands from the bulk toward the entire spectrum. More precisely, we prove that eigenvectors associated with energies $E$ satisfying $2 - |E| \gg N^{-c_{d,\alpha}}$ are delocalized, where the exponent $c_{d,\alpha}$ is given by $c_{d,\alpha} = 2\alpha - 1$ in dimension 1 and $c_{d,\alpha} = \alpha$ in dimension 2. Furthermore, when $\alpha > 1-d/6$, all eigenvectors of $H$ become delocalized. We further establish quantum unique ergodicity for delocalized eigenvectors, as well as a rigidity estimate for the eigenvalues. Our findings extend previous results -- established in the bulk regime for one-dimensional (1D) (arXiv:2501.01718) and two-dimensional (2D) (arXiv:2503.07606) random band matrices -- to the entire spectrum, including the spectral edges. They also complement the results of arXiv:0906.4047 and arXiv:2401.00492, which concern the edge eigenvalue statistics for 1D and 2D random band matrices.