### 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-1. We are interested in the crossover from straight-chain to random-walk behavior in the neighborhood of the point p=0, N-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.

Original language | English (US) |
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Pages (from-to) | 293-298 |

Number of pages | 6 |

Journal | Physical Review B |

Volume | 31 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1985 |

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## Cite this

*Physical Review B*,

*31*(1), 293-298. https://doi.org/10.1103/PhysRevB.31.293