Competition in growth and urns

TL;DR

This paper investigates competition dynamics in growth models and urn systems, proving that only one type survives almost surely across various settings, with implications for percolation and infection spread.

Contribution

It introduces a unified analysis showing that in all connected graphs, only one competing type persists, extending to growth models and urn systems with graph interactions.

Findings

Only one colour survives almost surely in the models.

In the 2D growth model, the other colour infects finitely many sites.

Results hold for all connected graphs and initial configurations.

Abstract

We study survival among two competing types in two settings: a planar growth model related to two-neighbour bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncoloured sites are given a colour at rate $0$, $1$ or $\infty$, depending on whether they have zero, one, or at least two neighbours of that colour. In the urn scheme, each vertex of a graph $G$ has an associated urn containing some number of either blue or red balls (but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same colour is added to each neighbouring urn, and balls in the same urn but of different colours annihilate on a one-for-one basis. We show that, for every connected graph $G$ and every initial configuration, only one colour survives almost surely. As a corollary, we deduce that in the two-type growth model on $\mathbb{Z}^2$, one of the colours only infects a finite number of sites with probability one. We also discuss generalisations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.