On the localization transition in three dimensions: Monte-Carlo simulation of a non-linear $\sigma$-model

We present a combination of analytical and numerical calculations for the critical behavior of a supersymmetric non-linear $\sigma$-model within the context of the localization transition of a disordered one-electron system. As a result, we obtain a localization length exponent and a set of inverse participation numbers in three dimensions. We find a continuous phase transition with the features of one-parameter scaling and multifractality at the critical point.