We consider Hermitian and symmetric random band matrices H in d ≥ dimensions. The matrix elements H_(xy), indexed by x,y ∈ Λ ? ?~d are independent and their variances satisfy σ_(xy)~2:= E{pipe}H_(xy){pipe}~2 = W~(-d) f((x-y)/W for some probability density 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 ? W~(d/3). We also show that the localization length of the eigenvectors of H is larger than a factor W~(d/6) times the band width W. All results are uniform in the size {pipe}Λ{pipe} of the matrix. This extends our recent result (Erdo{double acute}s and Knowles in Commun. Math. Phys., 2011) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying Σ_x σ_(xy)~2 for all y, the largest eigenvalue of H is bounded with high probability by 2+M~(-2/3+e){open} for any e{open} > 0, where M:= 1/(max_(x,y) σ_(xy)~2).
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