## Abstract

We study self-avoiding walks (SAWs) on non-Euclidean lattices that correspond to regular tilings of the hyperbolic plane ('hyperbolic graphs'). We prove that on all but at most eight such graphs, (i) there are exponentially fewer N-step self-avoiding polygons than there are N-step SAWs, (ii) the number of N-step SAWs grows as μ_{w}^{N} within a constant factor, and (iii) the average end-to-end distance of an N-step SAW is approximately proportional to N. In terms of critical exponents from statistical physics, (ii) says that γ = 1 and (iii) says that v = 1. We also prove that γ is finite on all hyperbolic graphs, and we prove a general identity about non-reversing walks that had previously been discovered for certain special cases.

Original language | English (US) |
---|---|

Pages (from-to) | 523-548 |

Number of pages | 26 |

Journal | Combinatorics Probability and Computing |

Volume | 14 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2005 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics