Critical behaviour of correlated and anisotropic self-avoiding walks

Abstract

The authors consider self-avoiding walks (SAWS) on a square lattice for which the choice of direction at each step is not entirely random, as in the case of self-avoiding random walks. In the case of correlated SAWS, the choice of the direction (consistent with the self-avoiding restriction) for the nth step is dependent on that for the (n-1)th step, while for the anisotropic SAWS, the probability to choose, at any step, the particular anisotropic lattice direction is different from that for the other directions. Both the extrapolation of exact enumeration results and a small cell real space renormalisation group study indicate that finite correlation does not affect the random SAW critical behaviour, while any finite amount of lattice anisotropy induce a crossover to the 'directed' SAW critical behaviour.