Competition between growths governed by Bernoulli Percolation

TL;DR

This paper investigates a competitive growth model on a lattice driven by Bernoulli percolation, showing that coexistence of two infections is highly restricted, occurring only at countably many parameter values.

Contribution

It establishes a rigorous result on the limited parameter regimes allowing coexistence in a Bernoulli percolation-based competition model.

Findings

Coexistence occurs only at countably many parameter values.

For any fixed q, p must be within a specific range for coexistence.

The model extends understanding of competitive growth in percolation contexts.

Abstract

We study a competition model on $\mathbb{Z}^d$ where the two infections are driven by supercritical Bernoulli percolations with distinct parameters $p$ and $q$. We prove that, for any $q$, there exist at most countably many values of $p<\min(q, \overrightarrow{p\_c})$ such that coexistence can occur.