Lattice paths: vicious walkers and friendly walkers

In an earlier paper [4] the problem of vicious random walkers on a d-dimensional directed lattice was considered. \Vicious walkers" describes the situation in which two or more walkers arriving at the same lattice site annihilate one another. Accordingly, the only allowed con gurations are those in which contacts are forbidden. Alternatively expressed as a static rather than dynamic problem, vicious walkers are mutually self-avoiding networks of directed lattice walks, that is directed lattice paths, which in turn model directed polymer networks. The problem of vicious walkers was introduced to the mathematical physics literature by Fisher [5] in 1984, who also discussed a number of physical applications. The general model is one of P random walkers on a d-dimensional lattice, who at regular time intervals simultaneously take one step with equal probability in the direction of one of the allowed lattice vectors. In the combinatorics literature, an equivalent problem for any planar graph was treated by Gessel and Viennot [7], though some of their results had been anticipated in other settings [16, 15]. In considering lattice path problems in which the Gessel-Viennot formulation does not apply, Viennot [23] suggested a model where the mutual avoidance constraint is relaxed to the extent that two paths may share a site, and may even stay together for just one step, but must then diverge { though they may subsequently touch. We now call this model \the 2-friendlywalk model" since the number of consecutive sites the walkers are allowed to share, here two, distinguishes this model from vicious walkers, which are not allowed to share any consecutive sites. It also distinguishes it from the so-called osculating walkers, which form directed lattice paths that can share one, but no more than one, consecutive sites. This naming convention also leads us to make further natural generalisations, given below. Interestingly, each of the vicious, osculating and 2-friendly-walk models can be formulated as lattice statistical mechanical vertex models, which are models of ferroelectric materials (see