Diffusion Profile for Random Band Matrices: a Short Proof

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 $d\ge 1$. 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 $d\ge 2$, 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.