A unifying model of generalised random walks

Abstract

The authors propose a unifying model for generalised static random walks, in which each walk has a weight exp(-g Sigma (ni)alpha ), where ni is the occupation number of a site and alpha and g are variable parameters. Special cases of this general walk include the ordinary (Brownian) random walk, the self-avoiding walk, the lattice Domb-Joyce model (1972) and the interacting random walk model recently introduced by Stanley et al. (1983). The asymptotic properties of the walk are studied in one dimension by effective medium arguments and exact enumeration methods. For repulsive correlations the authors find SAW behaviour, while for attractive correlations the model is self trapping for 1< alpha <or=2 and exhibits anomalous diffusion, with continuously varying exponents, for 0<or= alpha <1. In higher dimensions comments are made about effective medium predictions, and their relationship to other generalised random walk models.