Nonmonotonic coexistence regions for the two-type Richardson model

TL;DR

This paper explores the conditions under which two competing infections can coexist on various graphs, revealing surprising parameter regions where coexistence occurs even when infection rates are unequal.

Contribution

It provides new examples of graphs with nonmonotonic coexistence regions, challenging previous conjectures about equal infection rates being necessary for coexistence.

Findings

Coexistence can occur at unequal infection rate ratios on certain graphs.

Some graphs exhibit nonmonotonic coexistence regions, with coexistence disappearing as rates become equal.

The set of parameters allowing coexistence can be more complex than previously thought.

Abstract

In the two-type Richardson model on a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, 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 $\mathcal{G}$ 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 $\mathbb{Z}^d$, $d\geq 2$, 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 $1$.