The Annals of Applied Probability

Coexistence for Richardson type competing spatial growth models

Christopher Hoffman

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We study a large family of competing spatial growth models. In these models the vertices in ℤd can take on three possible states {0,1,2}. Vertices in states 1 and 2 remain in their states forever, while vertices in state 0, which are adjacent to a vertex in state 1 (or state 2), can switch to state 1 (or state 2). We think of the vertices in states 1 and 2 as infected with one of two infections, while the vertices in state 0 are considered uninfected. In this way these models are variants of the Richardson model. We start the models with a single vertex in state 1 and a single vertex in state 2. We show that with positive probability state 1 reaches an infinite number of vertices and state 2 also reaches an infinite number of vertices. This extends results and proves a conjecture of Häggström and Pemantle [J. Appl. Probab. 35 (1998) 683–692]. The key tool is applying the ergodic theorem to stationary first passage percolation.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 739-747.

First available in Project Euclid: 1 February 2005

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Zentralblatt MATH identifier

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

First passage percolation Richardson’s model competing growth


Hoffman, Christopher. Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 (2005), no. 1B, 739--747. doi:10.1214/105051604000000729.

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