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

TL;DR

This paper develops a $T$-expansion for Green's functions of high-dimensional random band matrices, enabling proof of local laws and advancing understanding of delocalization and quantum diffusion in such matrices.

Contribution

It introduces a new $T$-expansion method for analyzing Green's functions of high-dimensional random band matrices, improving previous results on delocalization.

Findings

Constructed an $T$-expansion with error $O(W^{-nd/2})$

Proved a local law for Green's functions of high-dimensional band matrices

Supported delocalization and quantum diffusion in dimensions $d extgreater=8$

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.