## The Annals of Applied Probability

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

### Coexistence for Richardson type competing spatial growth models

#### Abstract

We study a large family of competing spatial growth models. In these models the vertices in ℤ^{d} 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

**Source**

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

**Dates**

First available in Project Euclid: 1 February 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1107271666

**Digital Object Identifier**

doi:10.1214/105051604000000729

**Mathematical Reviews number (MathSciNet)**

MR2114988

**Zentralblatt MATH identifier**

1067.60098

**Subjects**

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

Secondary: 82B43: Percolation [See also 60K35]

**Keywords**

First passage percolation Richardson’s model competing growth

#### Citation

Hoffman, Christopher. Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15 (2005), no. 1B, 739--747. doi:10.1214/105051604000000729. https://projecteuclid.org/euclid.aoap/1107271666