Repulsion of random and self-avoiding walks from excluded points and lines

Abstract

The authors investigate the repulsion of random walks (RWs) and self-avoiding walks (SAWs) induced by (a) excluding a single lattice point or (b) excluding all points on a half-line from - infinity to the origin. For SAWs, they use exact enumeration and Monte Carlo methods to study the asymptotic behaviour of the displacement away from the excluded set on three-, four- and five-dimensional hypercubic lattices. When the SAW begins one lattice site away from an excluded point along the x direction, the mean displacement after N steps, (xN) approaches a finite limit, at a power-law rate, as N to infinity . However, the distribution of projected displacements exhibits a residual asymmetry in the asymptotic limit, reflective of a long-range influence of the excluded point. This general pattern of behaviour also occurs for a purely random walk. For a SAW which starts one lattice spacing away from the axis of the excluded half-line, enumeration data suggest that the mean displacement along the axis diverges as N nu , with v=3/4 in two dimensions, but as Nz, with z approximately=0.35, in three dimensions. This unexpected behaviour appears to be corroborated by constant-fugacity Monte Carlo data. However, for the corresponding RW model, both the solution to the diffusion equation and a heuristic argument indicate that (xN) diverges as N1/2 in two dimensions, and as N1/2/ln N in three dimensions.