Physica A 191 (1992) 379-385 North-Holland PllMcA L!‘l Multifractality of random walks and localized excitations on linear random fractals H. Eduardo Roman I. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Institut fiir Theoretische Physik, Universitiit Hamburg, Jungiusstrasse 9, W -2000 Hamburg 36, Germany Random walks and localized excitations on topologically one-dimensional random fractals, such as paths generated by simple random walks in d-dimensional lattices, are studied. These simple systems display a reach scenario of multifractal behavior in both the probability distribution of the random walks and the localization properties of the wave-functions. zyxwvutsrqponmlkji 1. Introduction In recent years, it has been realized that many dynamical processes in random media exhibit a variety of new complex phenomena (see e.g. reviews in ref. [ 11). These processes are generally characterized by broad distribution functions which display unusual scaling behavior and cannot be understood by means of traditional approaches. The concept of multifractality introduced by Mandelbrot [2] has provided us with a simple scheme to describe such distributions and has thus become an essential tool for dealing with these multifractal phenomena. Recently, Bunde, Havlin and the author [3] have shown that random walks on random fractals (such as the incipient infinite percolation cluster) are characterized by logarithmically broad distribution functions reflecting an underlying multifractal structure. Specifically, they studied the distribution function pi(r, t) of the random walks, representing the probability density that the walker is found, at time t, in a site i of the fractal located within a spherical shell of width dr at the distance r from its starting point at zyxwvutsrqponmlkjihgfedcbaZ t = 0. In the following, we briefly summarize the main results presented in ref. [3], in the light of more recent developments [4,5]. To obtain a detailed description of pi(r, t), and therefore of its fluctuations, it is convenient to study the generalized moments defined as [3] P,(r, t>= j$ C py(r, t) , r zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED r-l (1.1) 0378-4371/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved