Strict convexity of the planar exponential first-passage shape
Give each nearest-neighbour edge $e$ of $\mathbb{Z}^2$ an independent exponential random variable $\tau_e$ of mean one. Let $T(x, y)$ be the minimum of $\sum_{e \in \gamma} \tau_e$ over lattice paths $\gamma$ from $x$ to $y$, and let $\mu(v) = \lim_{n \to \infty} T(0, \lfloor nv \rfloor)/n$ be the deterministic time constant, with unit ball $B_\mu = \{v \in \mathbb{R}^2 : \mu(v) \le 1\}$, the limit shape. Lalley proved strict convexity of the limit shape under two hypotheses, on the location of geodesics and on mean-zero fluctuation limits of passage times. For passage times with support in $[1, \infty)$ and an atom at $1$ of mass greater than the critical probability of oriented percolation on $\mathbb{Z}^2$, the limit shape has a flat edge (Durrett and Liggett; Marchand).
Problem. Is $B_\mu$ strictly convex? Equivalently, must $\mu(su + (1 - s)v) < 1$ hold for all distinct $u, v \in \partial B_\mu$ and every $0 < s < 1$? The exponential law is fixed, and no curvature or straightness hypothesis may be assumed.
Source: Auffinger, Damron and Hanson (2017), Question 4, for every continuous passage-time distribution. Stated for exponential weights on Z^2 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, first-passage percolation, limit shapes
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 = {American Mathematical Society},
address = {Providence, RI},
year = {2017},
doi = {10.1090/ulect/068},
eprint = {1511.03262},
archivePrefix = {arXiv},
}
@article{durrett1981,
author = {Durrett, Richard and Liggett, Thomas M.},
title = {The shape of the limit set in {R}ichardson's growth model},
journal = {Ann. Probab.},
volume = {9},
number = {2},
year = {1981},
pages = {186--193},
doi = {10.1214/aop/1176994460},
}
@article{marchand2002,
author = {Marchand, R{\'e}gine},
title = {Strict inequalities for the time constant in first passage percolation},
journal = {Ann. Appl. Probab.},
volume = {12},
number = {3},
year = {2002},
pages = {1001--1038},
doi = {10.1214/aoap/1031863179},
}
@article{lalley2003,
author = {Lalley, Steven P.},
title = {Strict convexity of the limit shape in first-passage percolation},
journal = {Electron. Commun. Probab.},
volume = {8},
year = {2003},
pages = {135--141},
doi = {10.1214/ECP.v8-1089},
}
@misc{aim-strict-convexity-fpp,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Strict convexity of the planar exponential first-passage shape}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/169-strict-convexity-fpp.md},
note = {Entry 169 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that the limit shape of undirected nearest-neighbor first-passage percolation on $\Z^2$ with independent exponential edge weights is strictly convex and has a $C^1$ boundary. This resolves the strict convexity and differentiability conjectures for the planar exponential model. More generally, we prove differentiability of the time-constant norm for every Gamma edge-weight law with positive shape and rate.
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-00725, "Strict convexity of the planar exponential first-passage shape", clopen.solutions.