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

We prove lower bounds on the localization length of eigenfunctions in the three-dimensional Anderson model at weak disorders. Our results are similar to those obtained by Schlag, Shubin and Wolff for dimensions one and two. We prove that with probability one, most eigenfunctions have localization lengths bounded from below by $O(\frac{\lambda^{-2}}{\log\frac1\lambda})$, where $\lambda$ is the disorder strength. This is achieved by time-dependent methods which generalize those developed by Erd\"os and Yau to the lattice and non-Gaussian case. In addition, we show that the macroscopic limit of the corresponding lattice random Schr\"odinger dynamics is governed by the linear Boltzmann equations.