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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 randomvariables 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 bulkeigenvalue 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 alsoprove the quantum unique ergodicity for the bulk eigenvectors of $H$ and asharp 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 eigenvectorentries of $H$ are of order ${\mathrm{O}}(W^{-5/2}L^{-d/2+5/2})$ with highprobability.

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