Scaling in biased random walks

Abstract

We discuss the scaling of the statistics of a biased random walk. In the well-known model, a walk has probability p per step of stepping away from the direction of its last step. For a walk of N steps, such chains have persistence lengths ${p}^{\mathrm{\ensuremath{-}}1}$. We are interested in the crossover from straight-chain to random-walk behavior in the neighborhood of the point p=0, ${N}^{\mathrm{\ensuremath{-}}1}$=0. A scaling function has been computed for the problem of a random walk which avoids immediate returns by Schroll et al. Here we consider the self-avoiding case. A simple argument suggests a scaling function with crossover exponent 1. We present a proof that the crossover exponent is indeed one and numerical results for the scaling function on the square lattice which are consistent with this value. We also present a spin model which is asymptotically equivalent to the biased self-avoiding walk on the square lattice.