# How Long are the Arms in DBM?

## TL;DR

This paper studies the growth patterns of structures in the dielectric-breakdown model and provides new mathematical estimates for their arm lengths in 2D and 3D spaces.

## Contribution

A new proof for Kesten’s result and a generalized estimate for DBM growth in 2D and 3D using a different approach.

## Key findings

- The estimate for DBM growth depends on the model parameter and matches the best known results for DLA.
- The new proof applies to both 2D and 3D spaces using a different method from Kesten’s original approach.

## 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 \documentclass[12pt]{minimal}
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