On the radial growth of ballistic aggregation and other aggregation models

TL;DR

This paper extends a method to bound the radial growth of various aggregation models on integer lattices, including DLA and ballistic models, establishing the fractal dimension of ballistic clusters as 2, confirming a physics conjecture.

Contribution

It generalizes Kesten's method to a broad class of aggregation models and proves the fractal dimension of ballistic aggregation clusters in two dimensions is 2.

Findings

Bound on radial growth for a class of aggregation models.

Fractal dimension of ballistic clusters in 2D is 2.

Confirmation of a long-standing physics conjecture.

Abstract

For a class of aggregation models on the integer lattice $\mathbb{Z}^d$, $d\geq 2$, in which clusters are formed by particles arriving one after the other and sticking irreversibly where they first hit the cluster, including the classical model of diffusion-limited aggregation (DLA), we study the growth of the clusters. We observe that a method of Kesten used to obtain an almost sure upper bound on the radial growth in the DLA model generalizes to a large class of such models. We use it in particular to prove such a bound for the so-called ballistic model, in which the arriving particles travel along straight lines. Our bound implies that the fractal dimension of ballistic aggregation clusters in $\mathbb{Z}^2$ is 2, which proves a long standing conjecture in the physics literature.