The Annals of Applied Probability

Coexistence in two-type first-passage percolation models

Olivier Garet and Régine Marchand

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Abstract

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.

Article information

Source
Ann. Appl. Probab., Volume 15, Number 1A (2005), 298-330.

Dates
First available in Project Euclid: 28 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1106922330

Digital Object Identifier
doi:10.1214/105051604000000503

Mathematical Reviews number (MathSciNet)
MR2115045

Zentralblatt MATH identifier
1080.60092

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
Percolation first-passage percolation chemical distance competing growth

Citation

Garet, Olivier; Marchand, Régine. Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15 (2005), no. 1A, 298--330. doi:10.1214/105051604000000503. https://projecteuclid.org/euclid.aoap/1106922330


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