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Diffusion Profile for Random Band Matrices: A Short Proof

  • Yukun HeEmail author
  • Matteo Marcozzi
Article

Abstract

Let H be a Hermitian random matrix whose entries \(H_{xy}\) are independent, centred random variables with variances \(S_{xy} = {\mathbb {E}}|H_{xy}|^2\), where \(x, y \in ({\mathbb {Z}}/L{\mathbb {Z}})^d\) and \(d \geqslant 1\). The variance \(S_{xy}\) is negligible if \(|x - y|\) is bigger than the band width W. For \( d = 1\) we prove that if \(L \ll W^{1 + \frac{2}{7}}\) then the eigenvectors of H are delocalized and that an averaged version of \(|G_{xy}(z)|^2\) exhibits a diffusive behaviour, where \( G(z) = (H-z)^{-1}\) is the resolvent of H. This improves the previous assumption \(L \ll W^{1 + \frac{1}{4}}\) of Erdős et al. (Commun Math Phys 323:367–416, 2013). In higher dimensions \(d \geqslant 2\), we obtain similar results that improve the corresponding ones from Erdős et al. (Commun Math Phys 323:367–416, 2013). Our results hold for general variance profiles \(S_{xy}\) and distributions of the entries \(H_{xy}\). The proof is considerably simpler and shorter than that of Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013), Erdős et al. (Commun Math Phys 323:367–416, 2013). It relies on a detailed Fourier space analysis combined with isotropic estimates for the fluctuating error terms. It is completely self-contained and avoids the intricate fluctuation averaging machinery from Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013).

Notes

Acknowledgements

We thank Antti Knowles, who motivated us to study this problem, for useful discussions and suggestions on the topic. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715539_RandMat) and the Swiss National Science Foundation.

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Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of ZürichZürichSwitzerland
  2. 2.Section of MathematicsUniversity of GenevaGenevaSwitzerland

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