Failure of single-parameter scaling of wave functions in Anderson localization

We show how to use properties of the vectors which are iterated in the transfer-matrix approach to Anderson localization, in order to generate the statistical distribution of electronic wave-function amplitudes at arbitrary distances from the origin of L3` disordered systems. For d51 our approach is shown to reproduce exact diagonalization results available in the literature. In d52, where strips of width L<64 sites were used, attempted fits of Gaussian ~log-normal! forms to the wave-function amplitude distributions result in effective localization lengths growing with distance, contrary to the prediction from single-parameter scaling theory. We also show that the distributions possess a negative skewness S, which is invariant under the usual histogramcollapse rescaling, and whose absolute value increases with distance. We find 0.15&2S&0.30 for the range of parameters used in our study.