Probability distributions in the scaling theory of localization

Probability distributions in disordered electronic systems are studied by a real-space scaling transformation. It is shown that the resistance distribution at the mobility edge is very broad. The resistivity distribution at the metallic side of the transition is also broad, so that the metallic resistivity is not a self-averaging quantity. The calculation is consistent with a single-parameter scaling for the distributions. Near the mobility edge the transformation works better when the dimensionality $d$ is close to 2, i.e., $d=2+\varepsilon (\varepsilon \ll 1)$. The results are then extrapolated to $d=3\mathrm{}$.