## Annals of Applied Probability

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

### Coexistence in two-type first-passage percolation models

Olivier Garet and Régine Marchand

#### Abstract

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on ℤ^{d} 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*∈ℤ^{d}, there is a strictly positive probability that {*z*∈ℤ^{d};*d*(*y*,*z*)<*d*(*x*,*z*)} and {*z*∈ℤ^{d};*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