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Delocalization and Diffusion Profile for Random Band Matrices

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Abstract

We consider Hermitian and symmetric random band matrices H = (h xy ) in \({d\,\geqslant\,1}\) dimensions. The matrix entries h xy , indexed by \({x,y \in (\mathbb{Z}/L\mathbb{Z})^d}\), are independent, centred random variables with variances \({s_{xy} = \mathbb{E} |h_{xy}|^2}\). We assume that s xy is negligible if |xy| 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 \({|{G_{xy}}|^2}\) of the resolvent \({G=G(z)=(H-z)^{-1}}\) is self-averaging and we compute \({\mathbb{E} |{G_{xy}}|^2}\). We show that, as \({L\to\infty}\) and \({W\gg L^{4/5}}\), the behaviour of \({\mathbb{E} |G_{xy}|^2}\) is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.

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Correspondence to Antti Knowles.

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Communicated by M. Aizenman

László Erdős: Partially supported by SFB-TR 12 Grant of the German Research Council.

Antti Knowles: Partially supported by NSF grant DMS-0757425.

Horng-Tzer Yau: Partially supported by NSF grants DMS-0804279 and Simons Investigator Award.

Jun Yin: Partially supported by NSF grants DMS-1001655 and DMS-1207961.

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Erdős, L., Knowles, A., Yau, HT. et al. Delocalization and Diffusion Profile for Random Band Matrices. Commun. Math. Phys. 323, 367–416 (2013). https://doi.org/10.1007/s00220-013-1773-3

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