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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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1{
2 "domain": "mathematics",
3 "topics": [
4 "probability",
5 "statistical-mechanics"
6 ],
7 "type": "proof",
8 "source": {
9 "key": "aimpl",
10 "name": "AIM Problem Lists",
11 "url": "http://aimpl.org/firstpercolation/5/",
12 "id": "firstpercolation 5.2",
13 "curator": "American Institute of Mathematics",
14 "poser": "Olle Häggström and Robin Pemantle; posed by Christopher Hoffman",
15 "status": "open",
16 "as_of": "2026-10-20",
17 "catalog": "0521c19bbd38"
18 },
19 "review": true,
20 "approval": "first-time"
21}