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 \(\mathbb {Z}^3\).