@article{EJP321,
author = {Maria Deijfen and Olle Haggstrom},
title = {Nonmonotonic Coexistence Regions for the Two-Type Richardson Model on Graphs},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {11},
year = {2006},
keywords = {Competing growth; graphs; coexistence},
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 $1$.},
pages = {no. 13, 331--344},
issn = {1083-6489},
doi = {10.1214/EJP.v11-321},
url = {http://ejp.ejpecp.org/article/view/321}}