Abstract
We study analytically the localization of random walks, electrons and vibrations in self-similar structures. We find that in all cases the relevant localization functions ⟨ψ(r)⟩ N (mean probability distribution of random walkers, mean amplitude of the electronic wave function and mean vibrational amplitude) at distance r from the localization centre depend crucially upon the number N of configurations considered in the averages: there exists a crossover distance, r × ∼ rc (N) u , where rc (N) decreases with dimension d and increases logarithmically with N. For fractons and electrons, u = 1, and the localization exponent changes from 1 below r × to d min above r ×. For random walks, u = 1–1/d w, and the exponent changes from dw /(dw −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. We find that the ‘typical’ average ⟨ln ψ⟩ corresponds to the case N = 1. Our numerical results for random walks on percolation clusters and linear fractals agree quantitatively with these predictions.