Diffusion Profile for Random Band Matrices: a Short Proof

Let $H$ be a Hermitian random matrix whose entries $H_{xy}$ are independent, centred random variables with variances $S_{xy}=\mathbb{E}\mathrm{\mid }{H}_{xy}{\mathrm{\mid }}^2$, where $x,y\in (\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^d$ and $d\ge 1$. The variance $S_{xy}$ is negligible if $\mathrm{\mid }x-y\mathrm{\mid }$ 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 $\mathrm{\mid }{G}_{xy}(z){\mathrm{\mid }}^2$ exhibits a diffusive behaviour, where $G(z)=(H-z{)}^{-1}$ is the resolvent of $H$. This improves the previous assumption $L \ll W^{1 + \frac{1}{4}}$byErd\stackrel{˝}{o}setal\mathrm{.}(2013)\mathrm{.}Inhigherdimensions$d \geq 2$, 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.