The Annals of Applied Probability

Coexistence in two-type first-passage percolation models

Olivier Garet and Régine Marchand

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

First available in Project Euclid: 28 January 2005

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Zentralblatt MATH identifier

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

Percolation first-passage percolation chemical distance competing growth


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.

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  • Barsky, D. J., Grimmett, G. R. and Newman, C. M. (1991). Percolation in half-spaces: Equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields 90 111--148.
  • Boivin, D. (1990). First passage percolation: The stationary case. Probab. Theory Related Fields 86 491--499.
  • Deijfen, M. and Häggström, O. (2003). The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model. Available at http://www.
  • Garet, O. and Marchand, R. (2003). Asymptotic shape for the chemical distance and first-passage percolation in random environment. Available at PR/0304144.
  • Grimmett, G. (1999). Percolation, 2nd ed. Springer, Berlin.
  • Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35 683--692.
  • Harris, T. E. (1960). A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56 13--20.
  • Kesten, H. (1986). Aspects of first passage percolation. École d'Été de Probabilités de Saint-Flour XIV---1984. Lecture Notes in Math. 1180 125--264. Springer, Berlin.
  • Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.