Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(\mathbb Z/L \mathbb Z)^d$, where the entries $h_{xy}=\overline h_{yx}$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variance matrix $S=(s_{xy})$ has a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For dimensions $d\ge 7$, we prove the bulk eigenvalue universality of $H$ under the condition $W \gg L^{95/(d+95)}$. Assuming that $W\geq L^\epsilon$ for a small constant $\epsilon >0$, we also prove the quantum unique ergodicity for the bulk eigenvectors of $H$ and a sharp local law for the Green's function $G(z)=(H-z)^{-1}$ up to ${\mathrm{Im}} \, z \gg W^{-5}L^{5-d}$. The local law implies that the bulk eigenvector entries of $H$ are of order ${\mathrm{O}}(W^{-5/2}L^{-d/2+5/2})$ with high probability.