Delocalization and diffusion profile for random band matrices

Laszlo Erdos University of Munich Antti Knowles New York University Horng-Tzer Yau Harvard University Jun Yin University of Wisconsin-Madison

Probability mathscidoc:1608.28014

Communications in Mathematical Physics, 323, (1), 367-416, 2013
We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \in (\bZ/L\bZ)^d$, are independent, centred random variables with variances $s_{xy} = \E |h_{xy}|^2$. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix entries $\abs{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging and we compute $\E \abs{G_{xy}}^2$. We show that, as $L\to\infty$ and $W\gg L^{4/5}$, the behaviour of $\E |G_{xy}|^2$ is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.
random band matrix, Anderson model, localization length, quantum di usion
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@inproceedings{laszlo2013delocalization,
  title={Delocalization and diffusion profile for random band matrices},
  author={Laszlo Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823162146940119411},
  booktitle={Communications in Mathematical Physics},
  volume={323},
  number={1},
  pages={367-416},
  year={2013},
}
Laszlo Erdos, Antti Knowles, Horng-Tzer Yau, and Jun Yin. Delocalization and diffusion profile for random band matrices. 2013. Vol. 323. In Communications in Mathematical Physics. pp.367-416. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160823162146940119411.
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