Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices H in d ≥ 1 dimensions. The matrix elements Hxy, indexed by $${x,y \in \Lambda \subset \mathbb{Z}^d}$$, are independent, uniformly distributed random variables if $${\lvert{x-y}\rvert}$$ is less than the band width W, and zero otherwise. 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 Wd/6 times the band width. All results are uniform in the size $${\lvert{\Lambda}\rvert}$$ of the matrix.