Stabilization of DLA in a wedge

TL;DR

This paper proves that in a wedge with an angle less than π/4, the Diffusion Limited Aggregation process stabilizes within growing balls, enabling a finite-time construction of the infinite DLA in such geometries.

Contribution

It establishes the stabilization of DLA in wedges with angles smaller than π/4, providing a rigorous foundation for defining infinite DLA in these domains.

Findings

DLA stabilizes in wedges with angle < π/4

All new particles eventually move beyond any fixed radius

Enables finite-time construction of infinite DLA in wedges

Abstract

We consider Diffusion Limited Aggregation (DLA) in a two-dimensional wedge. We prove that if the angle of the wedge is smaller than $\pi/4$, there is some $a>2$ such that almost surely, for all $R$ large enough, after time $R^a$ all new particles attached to the DLA will be at distance larger than $R$ from the origin. This means that DLA stabilizes in growing balls, thus allowing a definition of the infinite DLA in a wedge via a finite time process.