# Coexistence in a random growth model with competition

## TL;DR

This paper studies a modified random growth model with two competing regions, analyzing conditions for their coexistence and the convergence of their harmonic measures to a non-trivial ergodic limit.

## Contribution

It introduces a variation of the Hastings-Levitov model with competition, exploring conditions for coexistence and harmonic measure convergence.

## Key findings

- Conditions for coexistence of regions identified
- Harmonic measures can converge to a non-trivial ergodic limit
- Model variation captures competitive growth dynamics

## Abstract

We consider a variation of the Hastings-Levitov model HL(0) for random growth in which the growing cluster consists of two competing regions. We allow the size of successive particles to depend both on the region in which the particle is attached, and the harmonic measure carried by that region. We identify conditions under which one can ensure coexistence of both regions. In particular, we consider whether it is possible for the process giving the relative harmonic measures of the regions to converge to a non-trivial ergodic limit.