On backbends on percolation backbones

Abstract

Just above the percolation concentration, a path on the backbone which leads from one side of the lattice to the other is not direct by zigzags through the lattice. Backbends are the portions of the zigzags which go backwards. They are important in the problem of particle transport in strong external fields, as they act as traps and limit the current. The threshold concentration for the proliferation of paths with backbends no longer than a given length L is defined as p_b(L), with the limits p_b(0)=p_d (directed percolation) and p_b(∞)=p_c (ordinary percolation). The inverse function ζ(p) is the smallest integer such that, for given p between p_c and p_d, there are paths to infinity on which every backbend is smaller than zeta (p). This minimal backbend length is computed on a Bethe lattice and shown to diverge as (p-p_c)^-½/. It is argued heuristically that on all lattices ζ(p) is proportional to the correlation length in the limit p → p_c. The chemical lengths of minimal backbend paths on the Bethe lattice are calculated.