LOCALIZATION IN DISORDERED STRUCTURES - BREAKDOWN OF THE SELF-AVERAGING HYPOTHESIS

We find that the relevant quantities describing the localization of electrons, vibrations, and random walks on random fractals are non-self-averaging. There exists a crossover distance r(x) that increases logarithmically with the number N of configurations considered in the averages. For vibrations and electrons, the localization exponent changes from 1 below r(x) to d(min) above r(x). For random walks, the exponent changes from d(u)/(d(u) - 1) to d(min)d(w)/(d(w)-d(min)), where d(w) and d(min) are the fractal dimensions of the random walk and the shortest path on the fractal, respectively. Our results explain the controversies regarding the localization exponent.