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 $$S_{xy} = {\mathbb {E}}|H_{xy}|^2$$, 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 Poincare 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 Poincare 14:1837–1925, 2013).