Distribution of shortest path lengths in percolation on a hierarchical lattice

Some characteristics of the shortest paths connecting distant points on a percolation network are studied. Attention is focused on the probability distribution of the lengths of shortest paths, especially on the manner in which this distribution changes as the distance between the two points is increased. Calculations are performed on a bond-diluted hierarchical lattice of the Wheatstone-bridge type. The evolution of the probability distribution is followed numerically. At the critical percolation concentration, the distribution is seen to approach a nontrivial function under proper rescaling of its argument. Away from criticality, the scaled distribution approaches a δ function whose location 1/v is a measure of how tortuous the shortest paths are. Here v is the wetting velocity. As the bond occupation probability is increased, there is a second phase transition when the shortest paths coincide with directed paths whose lengths are the smallest possible (v=1). This occurs at the directed percolation concentration. It is conjectured that the variation of v near the directed threshold is governed by the exponent ${\nu }_{?}$ which describes the divergence of the parallel correlation length in directed percolation.