Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices H in $${d \geqslant 1}$$ dimensions. The matrix elements Hxy, indexed by $${x,y \in \Lambda \subset \mathbb{Z}^d}$$, are independent and their variances satisfy $${\sigma_{xy}^2:=\mathbb{E} |{H_{xy}}|^2 = W^{-d} f((x - y)/W)}$$ for some probability density f. We assume that the law of each matrix element Hxy 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 $${t\ll W^{d/3}}$$ . 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 |Λ| of the matrix. This extends our recent result (Erdős 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 $${\sum_x\sigma_{xy}^2=1}$$ for all y, the largest eigenvalue of H is bounded with high probability by $${2 + M^{-2/3 + \varepsilon}}$$ for any $${\varepsilon > 0}$$, where $${M := 1 / (\max_{x,y}\sigma_{xy}^2)}$$ .