Backbends in Directed Percolation

When directed percolation in a bond percolation process does not occur, any path to infinity on the open bonds will zigzag back and forth through the lattice. Backbends are the portions of the zigzags that go against the percolation direction. They are important in the physical problem of particle transport in random media in the presence of a field, as they act to limit particle flow through the medium. The critical probability for percolation along directed paths with backbends no longer than a given length n is defined as pn. We prove that (pn) is strictly decreasing and converges to the critical probability for undirected percolation pc. We also investigate some variants of the basic model, such as by replacing the standard d-dimensional cubic lattice with a (d−1)-dimensional slab or with a Bethe lattice; and we discuss the mathematical consequences of alternative ways to formalize the physical concepts of “percolation” and “backbend.”