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

Fan Yang University of Pennsylvania Horng-Tzer Yau Harvard University Jun Yin University of California, Los Angeles

Probability mathscidoc:2110.28002

2021.10
We consider Green's functions $G(z):=(H-z)^{-1}$ of Hermitian random band matrices $H$ on the $d$-dimensional lattice $(\Z/L\Z)^d$. The entries $h_{xy}=\overline h_{yx}$ of $H$ are independent centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$, which satisfy a banded profile so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. For any fixed $n\in \N$, we construct an expansion of the $T$-variable, $T_{xy}=|m|^2 \sum_{\alpha}s_{x\alpha}|G_{\alpha y}|^2$, with an error $\OO(W^{-nd/2})$, 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 \cite{PartI_high} of this series.
Random band matrices, delocalization, quantum diffusion
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@inproceedings{fan2021delocalization,
  title={Delocalization and quantum diffusion of random band matrices in high dimensions II: T-expansion},
  author={Fan Yang, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20211001122013828340873},
  year={2021},
}
Fan Yang, Horng-Tzer Yau, and Jun Yin. Delocalization and quantum diffusion of random band matrices in high dimensions II: T-expansion. 2021. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20211001122013828340873.
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