How Long are the Arms in DBM?

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mn>2</mml:mn></mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mn>2</mml:mn></mml:msup> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:math> .