Lattice paths:: vicious walkers and friendly walkers

We introduce a model of friendly walkers which generalises the well-known vicious walker model. Friendly walkers refers to a model in which any number P of directed lattice paths, starting at adjacent lattice sites, simultaneously proceed in one of the allowed lattice directions. In the case of n-friendly walkers the paths may stay together for n vertices. The previously considered case of vicious walkers corresponds to the case n = 0. The Gessel-Viennot theorem applies only to vicious walkers, and not to the cases n > 0. The connection between this model and the m-vertex models of Statistical Mechanics is described. For planar configurations, we solve the two-walker case for all n. Conjectured solutions for the three-walker case with n = 1 are also obtained. Numerical studies lead to the conjectured asymptotic behaviour for all n, for an arbitrary number of walkers P and in arbitrary spatial dimension d greater than or equal to 2. (C) 2002 Elsevier Science B.V. All rights reserved.