# Coexistence in two-type first-passage percolation models

Abstract : We study the problem of coexistence in a two-type competition model governedby first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation.Actually, we prove for a large class of ergodic stationary passage times that for distinct points $x,y\in\Zd$, there is a strictly positive probability that$\{z\in\Zd;d(y,z)d(x,z)\}$ are both infinite sets.We also show that there is a strictly positive probability that the graph of time-minimizing path from the origin in first-passage percolation has at least two topological ends. This generalizes results obtained by H{ä}ggstr{ö}m and Pemantle forindependent exponential times on the square lattice.
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Cited literature [8 references]

https://hal.archives-ouvertes.fr/hal-00000971
Contributor : Olivier Garet <>
Submitted on : Thursday, December 18, 2003 - 10:37:02 PM
Last modification on : Monday, February 18, 2019 - 7:52:04 PM
Long-term archiving on: Monday, March 29, 2010 - 5:16:04 PM

### Citation

Olivier Garet, Régine Marchand. Coexistence in two-type first-passage percolation models. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2005, 15, No. 1A, p. 298-330. ⟨hal-00000971⟩

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