## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 11 (2006), paper no. 13, 331-344.

### Nonmonotonic Coexistence Regions for the Two-Type Richardson Model on Graphs

Maria Deijfen and Olle Haggstrom

#### Abstract

In the two-type Richardson model on a graph $G=(V,E)$, each vertex is at a given time in state $0$, $1$ or $2$. A $0$ flips to a $1$ (resp.\ $2$) at rate $\lambda_1$ ($\lambda_2$) times the number of neighboring $1$'s ($2$'s), while $1$'s and $2$'s never flip. When $G$ is infinite, the main question is whether, starting from a single $1$ and a single $2$, with positive probability we will see both types of infection reach infinitely many sites. This has previously been studied on the $d$-dimensional cubic lattice $Z^d$, $d\geq 2$, where the conjecture (on which a good deal of progress has been made) is that such coexistence has positive probability if and only if $\lambda_1=\lambda_2$. In the present paper examples are given of other graphs where the set of points in the parameter space which admit such coexistence has a more surprising form. In particular, there exist graphs exhibiting coexistence at some value of $\frac{\lambda_1}{\lambda_2} \neq 1$ and non-coexistence when this ratio is brought closer to $1$.

#### Article information

**Source**

Electron. J. Probab., Volume 11 (2006), paper no. 13, 331-344.

**Dates**

Accepted: 8 May 2006

First available in Project Euclid: 31 May 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1464730548

**Digital Object Identifier**

doi:10.1214/EJP.v11-321

**Mathematical Reviews number (MathSciNet)**

MR2217820

**Zentralblatt MATH identifier**

1113.60094

**Subjects**

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

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

**Keywords**

Competing growth graphs coexistence

**Rights**

This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Deijfen, Maria; Haggstrom, Olle. Nonmonotonic Coexistence Regions for the Two-Type Richardson Model on Graphs. Electron. J. Probab. 11 (2006), paper no. 13, 331--344. doi:10.1214/EJP.v11-321. https://projecteuclid.org/euclid.ejp/1464730548