### Abstract

An H-geodesic is a doubly infinite path which locally minimizes the passage time in the i.i.d. first passage percolation model on a half-plane H. Under the assumption that the bond passage times are continuously distributed with a finite mean, we prove that, with probability 1, H-geodesics do not exist. As a corollary we show that, with probability 1, any geodesic in the analogous model on the whole plane Z^{2} has to intersect all straight lines with rational slopes.

Original language | English (US) |
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Pages (from-to) | 358-367 |

Number of pages | 10 |

Journal | Annals of Probability |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1998 |

### Keywords

- Ergodicity
- First-passage percolation
- Infinite geodesics
- Large deviation bounds
- Time-minimizing paths

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

Wehr, J., & Woo, J. (1998). Absence of geodesics in first-passage percolation on a half-plane.

*Annals of Probability*,*26*(1), 358-367. https://doi.org/10.1214/aop/1022855423