Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in d=3

We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are analogous to those obtained by Shubin, Schlag and Wolff for dimensions one and two. We show that the localization length is, with probability one, bounded from below by $\sim\lambda^{-2} /|\log\lambda|$ for most eigenfunctions, where $\lambda$ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erdos and Yau to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lattice random Schrodinger dynamics is governed by the linear Boltzmann equations.