On the Radial Growth of Ballistic Aggregation and Other Aggregation Models

For a class of aggregation models on the integer lattice \({{\mathbb {Z}}}^d\) ${\mathbb{Z}}^d$ , \(d\ge 2\) $d\ge 2$ , 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 \({{\mathbb {Z}}}^2\) ${\mathbb{Z}}^2$ is 2, which proves a long standing conjecture in the physics literature.