Absence of doubly infinite minimizing paths in planar first-passage percolation
Assign independent exponential passage times $\tau_e$ of mean one to the nearest-neighbour edges $e$ of $\mathbb{Z}^2$, and let $T(x,y)$ be the least total passage time of a path from $x$ to $y$. A bigeodesic is an injective nearest-neighbour path $(v_k)_{k \in \mathbb{Z}}$ such that $$\sum_{k=i}^{j-1} \tau_{\{v_k, v_{k+1}\}} = T(v_i, v_j) \qquad \text{for all integers } i < j.$$ For continuous passage times with finite mean, Wehr and Woo proved that first-passage percolation on the upper half-plane almost surely has no bigeodesics, and Wehr that the number of bigeodesics in the plane is almost surely constant and either $0$ or $\infty$. Damron and Hanson proved that there are no bigeodesics with one end directed in a given deterministic direction, assuming the boundary of the limit shape is differentiable.
Problem. Is the probability that a bigeodesic exists equal to zero? No restriction is placed on the directions of its ends, and both ends lie in the full plane.
Source: Furstenberg, as reported by Auffinger, Damron and Hanson (2017), §4.5, Question 31, for general passage times. Stated by AIM — Open Applied Problems for mean-one exponential passage times on the square lattice.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, bigeodesics, first-passage percolation
References
@book{auffinger2017,
author = {Auffinger, Antonio and Damron, Michael and Hanson, Jack},
title = {50 Years of First-Passage Percolation},
series = {University Lecture Series},
volume = {68},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2017},
doi = {10.1090/ulect/068},
eprint = {1511.03262},
archivePrefix = {arXiv},
}
@article{damron2017,
author = {Damron, Michael and Hanson, Jack},
title = {Bigeodesics in first-passage percolation},
journal = {Comm. Math. Phys.},
volume = {349},
number = {2},
year = {2017},
pages = {753--776},
doi = {10.1007/s00220-016-2743-3},
eprint = {1512.00804},
archivePrefix = {arXiv},
}
@misc{aim-absence-fpp-bigeodesics,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Absence of doubly infinite minimizing paths in planar first-passage percolation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/171-absence-fpp-bigeodesics.md},
note = {Entry 171 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that planar first-passage percolation with independent identically distributed nonnegative nonatomic edge weights has almost surely no doubly infinite geodesic, provided the minimum of four independent weights has finite second moment. This resolves the planar no-bigeodesics conjecture under that moment assumption. The conclusion rules out all bigeodesics simultaneously, without any regularity assumption on the limit shape.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00659, "Absence of doubly infinite minimizing paths in planar first-passage percolation", clopen.solutions.