Distribution of Fractal Dimensions at the Anderson Transition

We investigated numerically the distribution of participation numbers in the 3D Anderson tight-binding model at the localization-delocalization threshold. In a single system these numbers fluctuate strongly from level to level. The fluctuations grow with system size. We argue that this mainly results from the fluctuations of the correlation dimension D₂ of the wave functions, whose distribution at the transition is calculated. In the thermodynamic limit ( L→∞) it does not depend on the system size L. This limiting distribution vanishes exactly at D₂ₘₐₓ0ex0ex=0ex0ex1.8, the highest possible value of the correlation dimension at the Anderson threshold in this model.