Probability distribution of the shortest path on the percolation cluster, its backbone, and skeleton

We consider the mean distribution functions [Formula Presented] [Formula Presented] and [Formula Presented] giving the probability that two sites on the incipient percolation cluster, on its backbone and on its skeleton, respectively, connected by a shortest path of length l are separated by an Euclidean distance [Formula Presented] Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents [Formula Presented] and [Formula Presented] which determine the scaling behavior of the distribution functions in the limit [Formula Presented] i.e., [Formula Presented] [Formula Presented] and [Formula Presented] with [Formula Presented] where [Formula Presented] and [Formula Presented] are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for [Formula Presented] [Formula Presented] and [Formula Presented] are in very good agreement with our numerical results.