The two-type Richardson model with unbounded initial configurations

TL;DR

This paper investigates the two-type Richardson model with unbounded initial configurations on b^d, showing conditions under which both infection types can grow unboundedly, depending on initial setup and infection intensities.

Contribution

It extends the understanding of the Richardson model to unbounded initial configurations, identifying conditions for simultaneous unbounded growth of competing infections.

Findings

Type 2 can grow unboundedly if it has higher intensity with initial configuration on hyperplane.

Type 2 can also grow unboundedly with equal intensity if initial infection is on negative axis.

Positive probability of unbounded growth depends on initial configuration and infection intensities.

Abstract

The two-type Richardson model describes the growth of two competing infections on $\mathbb{Z}^d$ and the main question is whether both infection types can simultaneously grow to occupy infinite parts of $\mathbb{Z}^d$. For bounded initial configurations, this has been thoroughly studied. In this paper, an unbounded initial configuration consisting of points $x=(x_1,...,x_d)$ in the hyperplane $\mathcal{H}=\{x\in\mathbb{Z}^d:x_1=0\}$ is considered. It is shown that, starting from a configuration where all points in $\mathcal{H} {\mathbf{0}\}$ are type 1 infected and the origin $\mathbf{0}$ is type 2 infected, there is a positive probability for the type 2 infection to grow unboundedly if and only if it has a strictly larger intensity than the type 1 infection. If, instead, the initial type 1 infection is restricted to the negative $x_1$-axis, it is shown that the type 2 infection at the origin can also grow unboundedly when the infection types have the same intensity.