Exact enumeration of parallel walks on directed lattices

Abstract

Mutually avoiding pairs of parallel walks on a number of d-dimensional lattices are mapped onto undirected random walks which return to the origin on a projected (d-1)-dimensional lattice. Generating functions for the number of such pairs of a given length are thus expressed in terms of standard Green's functions. Directed lattices considered include the directed hypercubic and body-centred hypercubic. The generating function for a directed triangular lattice is also obtained. This work is a generalization of known results for the square lattice.