Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices H in d > 1 dimensions. The matrix elements Hxy, indexed by x, y ∈ Λ ⊂ Z, are independent and their variances satisfy σ xy := E|Hxy| = W−df((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 W . 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 [1] to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying ∑ x σ 2 xy = 1 for all y, the largest eigenvalue of H is bounded with high probability by 2 + M −2/3+ε for any ε > 0, where M := 1/(maxx,y σ 2 xy).