Shortest paths in percolation

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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 pc. Along the easy axes, nu is expected to vanish as (p-pc)theta near pc, while (1- nu ) is expected to vary as (pd-p)theta ' near the directed percolation concentration pd. The exponent theta is related to an exponent nu which characterises the shortest paths precisely at pc, and which has recently been determined numerically. The wetting velocity is calculated analytically on a randomly diluted Bethe lattice, and the values theta =1/2 and theta '=1 (with logarithmic corrections) are found. The direction dependence of nu is also investigated.

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