### 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) |
---|---|

Pages (from-to) | 358-367 |

Number of pages | 10 |

Journal | Annals of Probability |

Volume | 26 |

Issue number | 1 |

State | Published - Jan 1998 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*26*(1), 358-367.

**Absence of geodesics in first-passage percolation on a half-plane.** / Wehr, Jan; Woo, Jung.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 26, no. 1, pp. 358-367.

}

TY - JOUR

T1 - Absence of geodesics in first-passage percolation on a half-plane

AU - Wehr, Jan

AU - Woo, Jung

PY - 1998/1

Y1 - 1998/1

N2 - 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.

AB - 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.

KW - Ergodicity

KW - First-passage percolation

KW - Infinite geodesics

KW - Large deviation bounds

KW - Time-minimizing paths

UR - http://www.scopus.com/inward/record.url?scp=0347626543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347626543&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0347626543

VL - 26

SP - 358

EP - 367

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -