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

We consider the mean distribution functions Phi(r\l), Phi(B)(r\l), and Phi(S)(r\l), 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 r. Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents g(1)=d(f)+d(min)-d and g(1)(B)=g(1)(S)-3d(min)-d, which determine the scaling behavior of the distribution functions in the limit x=r/l(<(nu)over tilde>) much less than 1, i.e., Phi (r\l) proportional to l(-<(nu)over tilde>d)x(g1), Phi(B)(r\l) proportional to l(-<(nu)over tilde>d)x(g1B), and Phi(S)(r\l) proportional to l(-<(nu)over tilde>d)x(g1S), with <(nu)over tilde> = 1/d(min), where d(f) and d(min) are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for g(1), g(1)(B), and g(1)(S) are in very good agreement with our numerical results. [S1063-651X(98)50411-4].