The Annals of Probability

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

Jan Wehr and Jung Woo

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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 $\mathbf{Z}^2$ has to intersect all straight lines with rational slopes.

Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 358-367.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855423

Digital Object Identifier
doi:10.1214/aop/1022855423

Mathematical Reviews number (MathSciNet)
MR1617053

Zentralblatt MATH identifier
0937.60092

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
First-passage percolation time-minimizing paths infinite geodesics ergodicity large deviation bounds

Citation

Wehr, Jan; Woo, Jung. Absence of geodesics in first-passage percolation on a half-plane. Ann. Probab. 26 (1998), no. 1, 358--367. doi:10.1214/aop/1022855423. https://projecteuclid.org/euclid.aop/1022855423


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References

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  • TUCSON, ARIZONA 85721 E-MAIL: wehr@math.arizona.edu jwoo@math.arizona.edu