A toy model for DLA arm growth in a wedge

TL;DR

This paper introduces a simplified Markov chain model to study DLA arm growth in wedges, providing evidence for the conjecture of a single infinite arm in thin wedges through a bootstrapping growth argument.

Contribution

It presents a novel toy model for DLA arm growth in wedges and demonstrates the single-arm phenomenon in thin wedges using an iterative proof technique.

Findings

Confirmed the single-arm growth in thin wedges within the model

Developed a bootstrapping method to analyze growth rates

Provided insights into DLA behavior in constrained geometries

Abstract

In this paper, we consider a non-homogeneous discrete-time Markov chain which can be seen as a toy model for the growth of the arms of the DLA (Diffusion limited aggregation) process in a sub-linear wedge. It is conjectured that in a thin enough linear wedge there is only one infinite arm in the DLA cluster and we demonstrate this phenomenon in our model. The technique follows a bootstrapping argument, in which we iteratively prove ever faster growth rate.