Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices $H$ in $d\ge 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset \Z^d$,areindependentandtheirvariancessatisfy$\sigma_{xy}^2:=\E \abs{H_{xy}}^2 = W^{-d} f((x - y)/W)$forsomeprobabilitydensity$f$. We assume that the law of each matrix element $H_{xy}$ is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d\mathrm{/}3}$. We also show that the localization length of the eigenvectors of $H$ is larger than a factor $W^{d\mathrm{/}6}$ times the band width $W$. All results are uniform in the size $\text{abs}\mathrm{\Lambda }$ of the matrix. This extends our recent result \cite{erdosknowles} to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying ${\sum }_x{\sigma }_{xy}^2=1$ for all $y$, the largest eigenvalue of $H$ is bounded with high probability by $2+M^{-2\mathrm{/}3+ϵ}$ for any $ϵ>0$, where $M\text{deq}1\mathrm{/}({\max }_{x,y}{\sigma }_{xy}^2)$.