## Electronic Journal of Probability

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

#### 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

Rights

#### 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

#### References

• M. Deijfen and O. Häggström, The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model, Comb. Probab. Computing, to appear.
• M. Deijfen, O. Häggström, and J. Bagley, A stochastic model for competing growth on $\R^d$, Markov Proc. Relat. Fields 10 (2004), 217-248.
• O. Garet and R. Marchand, Coexistence in two-type first-passage percolation models, Ann. Appl. Probab 15 (2005), 298-330.
• O. Häggström and R. Pemantle, First passage percolation and a model for competing spatial growth, J. Appl. Probab. 35 (1998), 683-692.
• O. Häggström and R. Pemantle, Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model, Stoch. Proc. Appl 90 (2000), 207-222.
• C. Hoffman, Coexistence for Richardson type competing spatial growth models, Ann. Appl. Probab. 15 (2005), 739-747.
• F. Lundin, Omniparametric simulation of the two-type Richardson model, licentiate thesis 2003:6, Dept. of Mathematics, Chalmers and Gteborg University.