Scaling of the distribution of shortest paths in percolation

TL;DR

This paper proposes a scaling hypothesis for the distribution of shortest paths in percolation clusters, considering system size and site occupancy probability, and tests it in two-dimensional percolation.

Contribution

It introduces a new scaling hypothesis for shortest path distributions in percolation and validates it through two-dimensional percolation analysis.

Findings

The scaling hypothesis accurately describes shortest path distributions.

Finite size effects are incorporated into the scaling model.

Dependence on site occupancy probability is characterized.

Abstract

We present a scaling hypothesis for the distribution function of the shortest paths connecting any two points on a percolating cluster which accounts for {\it (i)} the effect of the finite size of the system, and {\it (ii)} the dependence of this distribution on the site occupancy probability $p$. We test the hypothesis for the case of two-dimensional percolation.