## Abstract

We prove that for the (Formula presented.) -regular tessellations of the hyperbolic plane by (Formula presented.) -gons, there are exponentially more self-avoiding walks of length (Formula presented.) than there are self-avoiding polygons of length (Formula presented.). We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed (Formula presented.), we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion (Formula presented.) as (Formula presented.); on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self-avoiding walks of length (Formula presented.) is comparable to the (Formula presented.) th power of their connective constant. Some of these results were previously obtained by Madras and Wu for all but finitely many regular tessellations of the hyperbolic plane.

Original language | English |
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Pages (from-to) | 435-473 |

Number of pages | 39 |

Journal | Journal of Graph Theory |

Volume | 106 |

Issue number | 3 |

Early online date | 19 Feb 2024 |

DOIs | |

Publication status | Published - 31 Jul 2024 |

### Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.### Funding

I would like to thank John Haslegrave and Agelos Georgakopoulos for their comments on a preliminary version of the current paper. I would also like to thank Tom Hutchcroft for useful discussions. This research is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639046).

Funders | Funder number |
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European Research Council | |

Horizon 2020 | 639046 |

Horizon 2020 |

## Keywords

- connective constant
- hyperbolic graph
- planar graph
- self-avoiding polygon
- self-avoiding walk

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Geometry and Topology