Delocalization and diffusion profile for random band matrices

We consider Hermitian and symmetric random band matrices H = (h xy ) in d⩾1 d ⩾ 1 dimensions. The matrix entries h xy , indexed by x,y∈(Z/LZ)d x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances sxy=E|hxy|2 s x y = E | h x y | 2 . We assume that s xy is negligible if |x − y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if W≫L4/5 W ≫ L 4 / 5 . We also show that the magnitude of the matrix entries |Gxy|2 | G x y | 2 of the resolvent G=G(z)=(H−z)−1 G = G ( z ) = ( H - z ) - 1 is self-averaging and we compute E|Gxy|2 E | G x y | 2 . We show that, as L→∞ L → ∞ and W≫L4/5 W ≫ L 4 / 5 , the behaviour of E|Gxy|2 E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.