## Abstract

Self-avoiding random walks (SAWs) are studied on several hierarchical lattices in a randomly disordered environment. An analytical method to determine whether their fractal dimension D_{saw} is affected by disorder is introduced. Using this method, it is found that for some lattices, D_{saw} is unaffected by weak disorder; while for others D_{saw} changes even for infinitestimal disorder. A weak disorder exponent λ is defined and calculated analytically [λ measures the dependence of the variance in the partition function (or in the effective fugacity per step)v∼L^{λ} on the end-to-end distance of the SAW, L]. For lattices which are stable against weak disorder (λ<0) a phase transition exists at a critical value v=v^{*} which separates weak- and strong-disorder phases. The geometrical properties which contribute to the value of λ are discussed.

Original language | English |
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Pages (from-to) | 147-167 |

Number of pages | 21 |

Journal | Journal of Statistical Physics |

Volume | 80 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1995 |

## Keywords

- Self-avoiding walks
- disordered environment
- fractals
- hierarchical lattices
- renormalization