Delocalization and Diffusion Profile for Random Band Matrices

We consider Hermitian and symmetric random band matrices $H=(h_{xy})$ in $d \geq 1$dimensions\mathrm{.}Thematrixentries$h_{xy}$,indexedby$x,y \in (\bZ/L\bZ)^d$,areindependent,centredrandomvariableswithvariances$s_{xy} = \E |h_{xy}|^2$\mathrm{.}Weassumethat$s_{xy}$isnegligibleif$|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4\mathrm{/}5}$. We also show that the magnitude of the matrix entries $\text{abs}{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z{)}^{-1}$ is self-averaging and we compute $\text{E}\text{abs}{G_{xy}}^2$. We show that, as $L\to \mathrm{\infty }$ and $W\gg L^{4/5}$,thebehaviourof$\E |G_{xy}|^2$ is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.