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

Abstract

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.

74 pages. This is a sequel of arXiv:2104.12048. Accepted by Communications in Mathematical Physics

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

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