Diffusion Profile for Random Band Matrices: a Short Proof

TL;DR

This paper provides a simplified and shorter proof demonstrating diffusive behavior and delocalization of eigenvectors for random band matrices, improving previous bounds and applying to general variance profiles.

Contribution

It offers a shorter, more straightforward proof of diffusive eigenvector behavior in random band matrices, extending results to higher dimensions and general variance profiles.

Findings

Eigenvectors are delocalized under certain conditions.

Diffusive behavior of the resolvent's squared entries.

Improved bounds on the relation between L and W.

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 $d \geq 1$. 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 Erd\H{o}s et al. (2013). In higher dimensions $d \geq 2$, we obtain similar results that improve the corresponding by Erd\H{o}s 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 Erd\H{o}s 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 Erd\H{o}s and collaborators.