Multi-colour competition with reinforcement

TL;DR

This paper investigates a multi-colour urn competition model on cycles, proving that with three or more types, only one survives almost surely, and explores related annihilative processes with geometric insights.

Contribution

It establishes that for three or more types on a cycle, coexistence is almost surely impossible, confirming a conjecture and analyzing the process's geometric structure.

Findings

Single survivor in multi-type competition on cycles

Graph structure influences coexistence for three or more types

Detailed description of an auto-annihilative process on the cycle

Abstract

We study a system of interacting urns where balls of different colour/type compete for their survival, and annihilate upon contact. For competition between two types, the underlying graph (finite and connected), determining the interaction between the urns, is known to be irrelevant for the possibility of coexistence, whereas for $K\ge3$ types the structure of the graph does affect the possibility of coexistence. We show that when the underlying graph is a cycle, competition between $K\ge3$ types almost surely has a single survivor, thus establishing a conjecture of Griffiths, Janson, Morris and the first author. Along the way, we give a detailed description of an auto-annihilative process on the cycle, which can be perceived as an expression of the geometry of a M\"obius strip in a discrete setting.