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 ∈ Λ ⊂ Z, are independent, uniformly distributed random variables if |x − y| 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 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. All results are uniform in the size |Λ| of the matrix.