One-dimensional long-range diffusion-limited aggregation III -- The limit aggregate

TL;DR

This paper investigates the structure of the limit aggregate in one-dimensional long-range diffusion-limited aggregation, revealing how walk moments influence renewal structure, density, and tree properties, and introducing a harmonic competition model.

Contribution

It provides new insights into the structure of the limit aggregate, including conditions for renewal structure and density, and introduces a novel harmonic competition model.

Findings

For walks with finite third moment, the limit aggregate has renewal structure and positive density.

For walks with finite variance, the renewal structure disappears and the aggregate has zero density.

The paper establishes a connection between the aggregate's tree structure and coexistence in the harmonic competition model.

Abstract

In this paper we study the structure of the limit aggregate $A_\infty = \bigcup_{n\geq 0} A_n$ of the one-dimensional long range diffusion limited aggregation process defined in [AABK09]. We show (under some regularity conditions) that for walks with finite third moment $A_\infty$ has renewal structure and positive density, while for walks with finite variance the renewal structure no longer exists and $A_\infty$ has 0 density. We define a tree structure on the aggregates and show some results on the degrees and number of ends of these random trees. We introduce a new "harmonic competition" model where different colours compete for harmonic measure, and show how the tree structure is related to coexistence in this model.