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

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.