2023/08/25 by Bosch, Tillmann, Winter, Steffen
#28A80 (Secondary) #60D05 #60J10 (Primary) #82B24 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2308.13193
For a class of aggregation models on the integer lattice ℤd, d≥ 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 ℤ2 is 2, which proves a long standing conjecture in the physics literature.