Result: 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 ∈ A C Zd, are independent and their variances satisfy σ2xy := E|Hxy|2 = 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 ≪ Wd/3. We also show that the localization length of the eigenvectors of H is larger than a factor Wd/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 ∑x σ2xy = 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 σ2xy).