The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model

TL;DR

This paper proves that in the two-type Richardson model, the initial infected sets do not influence the likelihood of both infections growing unboundedly, highlighting a fundamental property of the model's long-term behavior.

Contribution

It establishes that the initial configuration of infected sites does not affect the probability of mutual unbounded growth in the two-type Richardson model.

Findings

Initial sets are irrelevant for mutual unbounded growth probability.

Mutual unbounded growth depends on infection rates, not initial configuration.

The result holds for all initial finite infected sets.

Abstract

The two-type Richardson model describes the growth of two competing infections on $\mathbb{Z}^d$. At time 0 two disjoint finite sets $\xi_1,\xi_2\subset \mathbb{Z}^d$ are infected with type 1 and type 2 infection respectively. An uninfected site then becomes type 1 (2) infected at a rate proportional to the number of type 1 (2) infected nearest neighbors and once infected it remains so forever. The main result in this paper is, loosely speaking, that the choice of the initial sets $\xi_1$ and $\xi_2$ is irrelevant in deciding whether the event of mutual unbounded growth for the two infection types has positive probability or not.