Pair Connectedness and Shortest Path Scaling in Critical Percolation

TL;DR

This paper provides high-precision data on shortest path distributions in critical percolation, disproving a recent conjecture and offering bounds on cluster connectivity probabilities.

Contribution

It introduces an efficient algorithm for analyzing shortest paths and presents new empirical results that challenge previous theoretical conjectures in 2D percolation.

Findings

Disproves Porto et al.'s conjecture for 2D percolation

Provides upper bounds on probabilities of points being on different clusters

Introduces a new efficient algorithm for percolation analysis

Abstract

We present high statistics data on the distribution of shortest path lengths between two near-by points on the same cluster at the percolation threshold. Our data are based on a new and very efficient algorithm. For $d=2$ they clearly disprove a recent conjecture by M. Porto et al., Phys. Rev. {\bf E 58}, R5205 (1998). Our data also provide upper bounds on the probability that two near-by points are on different infinite clusters.