Structural and dynamical properties of the percolation backbone in two and three dimensions

We study structural and dynamical properties of the backbone of the incipient infinite cluster for site percolation in two and three dimensions. We calculate the average mass of the backbone in chemical [Formula Presented] space, [Formula Presented], where [Formula Presented] is the chemical dimension. We find [Formula Presented] in [Formula Presented] and [Formula Presented] in [Formula Presented]. The fractal dimension in [Formula Presented] space [Formula Presented] is obtained from the relation [Formula Presented] [Formula Presented] in [Formula Presented] and [Formula Presented] in [Formula Presented], where [Formula Presented] is the fractal dimension of the shortest path. The distribution function [Formula Presented] is determined, giving the probability of finding two backbone sites at the spatial distance [Formula Presented] connected by the shortest path of length [Formula Presented], as well as the related quantity [Formula Presented], giving the length of the minimal shortest path for two backbone sites at distance [Formula Presented] as a function of the number [Formula Presented] of configurations considered. Regarding dynamical properties, we study the distribution functions [Formula Presented] and [Formula Presented] of random walks on the backbone, giving the probability of finding a random walker after [Formula Presented] time steps, at a chemical distance [Formula Presented], and Euclidean distance [Formula Presented] from its starting point, respectively, and their first moments [Formula Presented] and [Formula Presented], from which the fractal dimensions of the random walk [Formula Presented] and [Formula Presented] are estimated. We find [Formula Presented] and [Formula Presented] in [Formula Presented] as well as [Formula Presented] and [Formula Presented] in [Formula Presented].