Delocalization of random band matrices at the edge

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.