Exclusion of coexistence for unequal Richardson growth rates
Consider the two-type Richardson model on $\mathbb{Z}^2$: a continuous-time process in which each site is vacant (state $0$) or occupied by type $1$ or type $2$. Initially only two distinct sites $x_1, x_2$ are occupied, with types $1$ and $2$ respectively. A vacant site becomes type $i$ at rate $\lambda_i$ times its number of nearest neighbours of type $i$, where $\lambda_1, \lambda_2 > 0$, and occupied sites never change type. Let $C_i$ be the set of sites that are eventually of type $i$. Häggström and Pemantle proved that for equal rates both types occupy infinitely many sites with positive probability, and that, with one rate fixed, this has probability zero for all but at most countably many values of the other. Ahlberg, Deijfen and Hoffman proved that in the upper half-plane, two types started on the horizontal axis coexist with positive probability if and only if their rates are equal.
Problem. Is $$\mathbb{P}\bigl(|C_1| = \infty,\ |C_2| = \infty\bigr) = 0$$ for every pair of rates $\lambda_1 \ne \lambda_2$? The question concerns every unequal pair of rates, not almost every ratio.
Source: Häggström and Pemantle (1998), conjecture in the introduction; restated by Auffinger, Damron and Hanson (2017), Question 34.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, competing growth, first-passage percolation, Richardson model
References
@article{haggstrom1998,
author = {H{\"a}ggstr{\"o}m, Olle and Pemantle, Robin},
title = {First passage percolation and a model for competing spatial growth},
journal = {J. Appl. Probab.},
volume = {35},
number = {3},
year = {1998},
pages = {683--692},
doi = {10.1239/jap/1032265216},
eprint = {math/9701226},
archivePrefix = {arXiv},
}
@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{haggstrom2000,
author = {H{\"a}ggstr{\"o}m, Olle and Pemantle, Robin},
title = {Absence of mutual unbounded growth for almost all parameter values in the two-type {R}ichardson model},
journal = {Stochastic Process. Appl.},
volume = {90},
number = {2},
year = {2000},
pages = {207--222},
doi = {10.1016/S0304-4149(00)00042-9},
}
@article{ahlberg2020,
author = {Ahlberg, Daniel and Deijfen, Maria and Hoffman, Christopher},
title = {The two-type {R}ichardson model in the half-plane},
journal = {Ann. Appl. Probab.},
volume = {30},
number = {5},
year = {2020},
pages = {2261--2273},
doi = {10.1214/19-AAP1557},
eprint = {1808.10796},
archivePrefix = {arXiv},
}
@misc{aim-unequal-richardson-competition,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Exclusion of coexistence for unequal Richardson growth rates}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/172-unequal-richardson-competition.md},
note = {Entry 172 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00486, "Exclusion of coexistence for unequal Richardson growth rates", clopen.solutions.