  We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band
matrices $H$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The
entries $h_{xy}=\bar h_{yx}$ of $H$ are independent centered complex Gaussian
random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variances
satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the
band width $W$. For any $n\in \mathbb N$, we construct an expansion of the
$T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, with an
error $O(W^{-nd/2})$, and use it to prove a local law on the Green's function.
This $T$-expansion was the main tool to prove the delocalization and quantum
diffusion of random band matrices for dimensions $d\ge 8$ in part I of this
series.
