The Häggström–Pemantle conjecture on competing growth

Prove that two competing growth processes on the lattice cannot both grow forever when their speeds differ.

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State the problem, from AIM Problem Lists

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	1	+# The Häggström–Pemantle conjecture on competing growth
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	3	+Two infections spread through the square grid, each claiming new sites from its own territory at its own rate. If both spread equally fast, they can both grow forever with positive probability. Häggström and Pemantle conjectured that if one is even slightly faster, then almost surely one of the two is eventually enclosed by the other and stops growing. This is known for all speed ratios except possibly countably many, but not for any specific ratio.
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	5	+## Problem
	6	+
	7	+In the two-type Richardson model on $\mathbb{Z}^d$ ($d \ge 2$), each site is empty or occupied by type 1 or type 2, and occupied sites stay occupied forever. An empty site becomes type $i$ at rate $\lambda_i$ times the number of its neighbours of type $i$, independently of everything else. Start with one site of each type, say at the origin and at a neighbouring site. Coexistence is the event that both types eventually occupy infinitely many sites.
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	9	+Conjecture (Häggström and Pemantle). If $\lambda_1 \neq \lambda_2$, then coexistence has probability zero.
	10	+
	11	+## Background
	12	+
	13	+Häggström and Pemantle (1998) introduced the model by starting Richardson's growth model from two sites of different colours, each newly infected site taking the colour of the site that infected it. With equal rates the model is a reformulation of first-passage percolation with exponential passage times, in which each site goes to the starting point that reaches it first, so coexistence is tied to the geometry of time-minimising paths. The conjecture is listed by Christopher Hoffman in the problem list of the AIM workshop on first passage percolation.
	14	+
	15	+## What is known
	16	+
	17	+- Equal rates: Häggström and Pemantle (1998) proved that coexistence has positive probability on $\mathbb{Z}^2$, and deduced that the tree of time-minimising paths from the origin in first-passage percolation with exponential passage times has at least two ends with positive probability. Garet and Marchand (2005) and Hoffman (2005) extended coexistence to $\mathbb{Z}^d$ for all $d \ge 2$ and to more general passage times.
	18	+- Häggström and Pemantle (2000) proved that, for every $d \ge 2$, coexistence has probability zero for all but at most countably many values of $\lambda_2 / \lambda_1$. As Deijfen and Häggström point out, this does not rule out coexistence for any given value.
	19	+- Garet and Marchand (2008) showed that with different speeds the two types cannot both occupy a positive density of space.
	20	+- Ahlberg, Deijfen and Hoffman (2020) proved the analogue in the upper half-plane $\mathbb{Z} \times \mathbb{Z}_{+}$: with types starting on the horizontal axis, coexistence has positive probability if and only if the two intensities are equal.
	21	+- On suitable graphs other than $\mathbb{Z}^d$, the set of rate ratios that allow coexistence can be an interval away from 1, or a finite set of points (Deijfen and Häggström, 2006), so arguments that ignore the geometry of the lattice cannot work.
	22	+
	23	+## What would count as progress
	24	+
	25	+- A proof of the conjecture for one explicit pair of different rates, which is not known for any pair.
	26	+- A proof that the probability of coexistence does not decrease as the ratio $\lambda_2 / \lambda_1 \le 1$ increases towards 1. This monotonicity conjecture of Deijfen and Häggström would imply the full conjecture when combined with the countable-exceptions theorem.
	27	+- Results for other geometries or initial conditions, such as strips, cones or half-spaces, that isolate where the full-lattice argument fails.
	28	+
	29	+## References
	30	+
	31	+1. O. Häggström and R. Pemantle, [First passage percolation and a model for competing spatial growth](https://doi.org/10.1239/jap/1032265216), J. Appl. Probab. 35 (1998), 683–692; [arXiv:math/9701226](https://arxiv.org/abs/math/9701226).
	32	+2. O. Häggström and R. Pemantle, [Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model](https://doi.org/10.1016/S0304-4149(00)00042-9), Stochastic Process. Appl. 90 (2000), 207–222.
	33	+3. O. Garet and R. Marchand, [Coexistence in two-type first-passage percolation models](https://doi.org/10.1214/105051604000000503), Ann. Appl. Probab. 15 (2005), 298–330.
	34	+4. C. Hoffman, [Coexistence for Richardson type competing spatial growth models](https://doi.org/10.1214/105051604000000729), Ann. Appl. Probab. 15 (2005), 739–747.
	35	+5. M. Deijfen and O. Häggström, [Nonmonotonic coexistence regions for the two-type Richardson model on graphs](https://doi.org/10.1214/EJP.v11-321), Electron. J. Probab. 11 (2006), 331–344.
	36	+6. M. Deijfen and O. Häggström, [The pleasures and pains of studying the two-type Richardson model](https://doi.org/10.1093/acprof:oso/9780199239252.003.0002), in Analysis and Stochastics of Growth Processes and Interface Models, Oxford University Press (2008), 39–54; [arXiv:1509.07006](https://arxiv.org/abs/1509.07006).
	37	+7. O. Garet and R. Marchand, [First-passage competition with different speeds: positive density for both species is impossible](https://doi.org/10.1214/EJP.v13-581), Electron. J. Probab. 13 (2008), 2118–2159.
	38	+8. D. Ahlberg, M. Deijfen and C. Hoffman, [The two-type Richardson model in the half-plane](https://doi.org/10.1214/19-AAP1557), Ann. Appl. Probab. 30 (2020), 2261–2273.
	39	+
	40	+## Source
	41	+
	42	+Imported from [AIM Problem Lists firstpercolation 5.2](http://aimpl.org/firstpercolation/5/), curated by American Institute of Mathematics. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.