Shortest paths in percolation

Abstract

The separation of two points on a percolation network is characterised not only by the distance between them, but also by the length of a path on the network which connects them. The wetting velocity nu provides a measure of the lengths of the shortest connecting paths on the network above the percolation concentration p_c. Along the easy axes, nu is expected to vanish as (p-p_c)^theta near p^c, while (1- nu ) is expected to vary as (p^d-p)^theta' near the directed percolation concentration p_d. The exponent theta is related to an exponent nu which characterises the shortest paths precisely at p_c, and which has recently been determined numerically. The wetting velocity is calculated analytically on a randomly diluted Bethe lattice, and the values theta =½ and theta'=1 (with logarithmic corrections) are found. The direction dependence of nu is also investigated.