Localization in disordered structures: Breakdown of the self-averaging hypothesis.

Abstract

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 ${\mathit{r}}_{\ifmmode\times\else\texttimes\fi{}}$ that increases logarithmically with the number N of configurations considered in the averages. For vibrations and electrons, the localization exponent changes from 1 below ${\mathit{r}}_{\ifmmode\times\else\texttimes\fi{}}$ to ${\mathit{d}}_{\mathrm{min}}$ above ${\mathit{r}}_{\ifmmode\times\else\texttimes\fi{}}$. For random walks, the exponent changes from ${\mathit{d}}_{\mathit{w}}$/(${\mathit{d}}_{\mathit{w}}$-1) to ${\mathit{d}}_{\mathrm{min}}$${\mathit{d}}_{\mathit{w}}$/(${\mathit{d}}_{\mathit{w}}$-${\mathit{d}}_{\mathrm{min}}$), where ${\mathit{d}}_{\mathit{w}}$ and ${\mathit{d}}_{\mathrm{min}}$ 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.