Delocalization and quantum diffusion of random band matrices in high dimensions II: $T$-expansion

We consider Green’s functions $G(z):=(H-z{)}^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^d$. The entries $h_{xy}={\bar{h}}_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb{E}\mathrm{\mid }{h}_{xy}{\mathrm{\mid }}^2$. The variances satisfy a banded profile so that $s_{xy}$ is negligible if $\mathrm{\mid }x-y\mathrm{\mid }$ exceeds the band width $W$. For any $n\in \mathbb{N}$, we construct an expansion of the $T$-variable, $T_{xy}=\mathrm{\mid }m{\mathrm{\mid }}^2{\sum }_{\alpha }{s}_{x\alpha }\mathrm{\mid }{G}_{\alpha y}{\mathrm{\mid }}^2$, with an error $O(W^{-nd\mathrm{/}2})$, 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.