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February 2005 Coexistence in two-type first-passage percolation models
Olivier Garet, Régine Marchand
Ann. Appl. Probab. 15(1A): 298-330 (February 2005). DOI: 10.1214/105051604000000503


We study the problem of coexistence in a two-type competition model governed by first-passage percolation on ℤd or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x,y∈ℤd, there is a strictly positive probability that {z∈ℤd;d(y,z)<d(x,z)} and {z∈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 Häggström and Pemantle for independent exponential times on the square lattice.


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Olivier Garet. Régine Marchand. "Coexistence in two-type first-passage percolation models." Ann. Appl. Probab. 15 (1A) 298 - 330, February 2005.


Published: February 2005
First available in Project Euclid: 28 January 2005

zbMATH: 1080.60092
MathSciNet: MR2115045
Digital Object Identifier: 10.1214/105051604000000503

Primary: 60K35 , 82B43

Keywords: Chemical distance , competing growth , First-passage percolation , percolation

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.15 • No. 1A • February 2005
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