Diffusion Profile for Random Band Matrices: a Short Proof

• Published in: arXiv.org 2019-09
• Main Authors: He, Yukun, Marcozzi, Matteo
• Format: Article
• Language: English
• Subjects: Eigenvectors; Independent variables; Random variables; Variance
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1804.09446

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 \(d \geq 1\). 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 \(L \ll W^{1 + \frac{1}{4}}\) by Erdős et al. (2013). In higher dimensions \(d \geq 2\), we obtain similar results that improve the corresponding by Erdős et al. 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 by Erdős et al. It relies on a detailed Fourier space analysis combined with isotropic estimates for the fluctuating error terms. It avoids the intricate fluctuation averaging machinery used by Erdős and collaborators.