How long are the arms in DBM?

TL;DR

This paper extends Kesten's estimate on the growth of Diffusion Limited Aggregation (DLA) to the Dielectric Breakdown Model (DBM) in two and three dimensions, providing new proofs and insights into their growth properties.

Contribution

It generalizes Kesten's growth estimate for DLA to the DBM, offering a novel approach and proof in both 2D and 3D settings.

Findings

Generalized growth estimates for DBM depending on its parameter

Provided new proofs for Kesten's estimate in 2D and 3D

Matched known results for DLA growth behavior

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 $\mathbb{Z}^3$.