Fractal Behavior of the Shortest Path Between Two Lines in Percolation Systems

TL;DR

This paper investigates the fractal nature of the shortest path between two lines in 3D percolation systems at criticality, revealing complex power law regimes influenced by line orientations.

Contribution

It introduces a scaling form for the shortest path distribution between lines with arbitrary positions and orientations in 3D percolation at criticality.

Findings

Up to four distinct power law regimes identified in the distribution

Exponents depend on the relative orientations of the lines

Scaling arguments explain the fractal behavior

Abstract

Using Monte-Carlo simulations, we determine the scaling form for the probability distribution of the shortest path, $\ell$, between two lines in a 3-dimensional percolation system at criticality; the two lines can have arbitrary positions, orientations and lengths. We find that the probability distributions can exhibit up to four distinct power law regimes (separated by cross-over regimes) with exponents depending on the relative orientations of the lines. We explain this rich fractal behavior with scaling arguments.