Random walk on self-avoiding walk: a model for the conductivity of linear polymers

Abstract

Random walks on self-avoiding walks (SAWs) are studied using Monte Carlo techniques on a square lattice (with nearest-neighbour hopping along the chain and between SAW points which are nearest neighbours on the embedding lattice). The average of the square of the end-to-end distance for random walks of t steps on SAWs of length N is fitted to the scaling forms (R_t²) varies as N^δ t^κ (for t < <N^θ ) and (R_t²) varies as N^{2 ν s} (for t > or approximately=N^θ), where theta approximately=2 ν s/k; nu s being the average end-to-end distance exponent for SAWs. The observed value of the exponent delta is supported by the authors' real space renormalisation group result for the conductivity of SAW chains. The exponent k has been related to the 'effective' fractal dimension of the SAW chain.