Pair connectedness and shortest-path scaling in critical percolation

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 1998 Phys. Rev. E 58 R5205. Our data also provide upper bounds on the probability that two near-by points, are on different infinite clusters.