How long are the arms in DBM?

Abstract

Diffusion Limited Aggregation and its generalization, Dielectric Breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten's estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in $\mathbb{Z}^2$ and $\mathbb{Z}^3$. The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten's, our argument provides a new proof for Kesten's result both in $\mathbb{Z}^2$ and $\mathbb{Z}^3$.