Probability Distribution of the Shortest Path on the Percolation Cluster, its Backbone and Skeleton

TL;DR

This paper derives and confirms analytical expressions for the probability distribution of Euclidean distances between sites connected by shortest paths on percolation clusters, backbone, and skeleton, revealing their scaling behavior.

Contribution

It introduces new analytical formulas for the distribution exponents related to shortest paths on percolation structures, supported by numerical validation.

Findings

Analytical expressions for distribution exponents g1, g1B, g1S derived.

Numerical results agree well with theoretical predictions.

Scaling behavior characterized for different percolation structures.

Abstract

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 g1=df+dmin-d and g1B=g1S-3dmin-d, which determine the scaling behavior of the distribution functions in the limit x=r/l^(nu) much less than 1, i.e., Phi(r|l) proportional to l^(-(nu)d)x^(g1), Phi(B)(r|l) proportional to l^(-(nu)d)x^(g1B), and Phi(S)(r|l) proportional to l^(-(nu)d)x^(g1S), with nu=1/dmin, where df and dmin are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for g1, g1B, and g1S are in very good agreement with our numerical results.