  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.
