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 $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}=\bar h_{yx}$ are independent centered complex Gaussian variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds width $W$. For any $n\in \mathbb N$, we construct an expansion $T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, error $O(W^{-nd/2})$, and use it to prove local law function. This $T$-expansion was main tool delocalization quantum diffusion for dimensions $d\ge 8$ in part I this series.