  We consider Hermitian random band matrices $H=(h_{xy})$ on the
$d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}$ are
independent (up to Hermitian conditions) centered complex Gaussian random
variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variance matrix
$S=(s_{xy})$ has a banded structure so that $s_{xy}$ is negligible if $|x-y|$
exceeds the band width $W$. In dimensions $d\ge 8$, we prove that, as long as
$W\ge L^\epsilon$ for a small constant $\epsilon>0$, with high probability most
bulk eigenvectors of $H$ are delocalized in the sense that their localization
lengths are comparable to $L$. Denote by $G(z)=(H-z)^{-1}$ the Green's function
of the band matrix. For ${\mathrm Im}\, z\gg W^2/L^2$, we also prove a widely
used criterion in physics for quantum diffusion of this model, namely, the
leading term in the Fourier transform of $\mathbb E|G_{xy}(z)|^2$ with respect
to $x-y$ is of the form $({\mathrm Im}\, z + a(p))^{-1}$ for some $a(p)$
quadratic in $p$, where $p$ is the Fourier variable. Our method is based on an
expansion of $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$ and it
requires a self-energy renormalization up to error $W^{-K}$ for any large
constant $K$ independent of $W$ and $L$. We expect that this method can be
extended to non-Gaussian band matrices.
