Quantum Diffusion and Delocalization for Band Matrices with General Distribution

Abstract

We consider Hermitian and symmetric random band matrices H in d ≥ dimensions. The matrix elements Hxy, indexed by x,y ∈ Λ ⊂ ℤd are independent and their variances satisfy σ2xy:= E{pipe}Hxy{pipe}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 ≪ 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 {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 Σx σ2xy 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/(maxx,y σ2xy). © 2011 Springer Basel AG.