Abstract
We study numerically and extensively the three-dimensional Anderson model for a disordered system on a simple-cubic lattice. Besides the mean values of the Lyapunov exponents, we calculate also their probability distributions in the metallic and the insulating regimes as well as near the critical point. We discuss the statistical properties of their differences. The universality properties of the distributions are investigated. From the statistical properties of the Lyapunov exponents, information about the statistics of the conductance is derived. The number of parameters, which determine the shape of the distribution of the conductance is one, one or two, and zero in the metallic and in the insulating phases, and at the critical point, respectively. It is shown that all of the distributions depend on the geometry of the sample. The results for the cubic geometry are compared with those obtained previously for the quasi-one-dimensional geometry using the one-parameter scaling hypothesis. The validity of the latter and the applicability of finite-size scaling in numerical studies of the Anderson transition are discussed.