How long are the arms in DBM?
Ilya Losev, Stanislav Smirnov
TL;DR
The paper addresses the growth of diffusion-limited aggregation and the dielectric-breakdown model on lattice spaces, extending Kesten's 2D/3D growth bounds for DLA to the DBM family with parameter $\eta$. It develops a tau-spectrum framework linking the growth rate to harmonic-measure multifractality via $D(\eta)=\tau(\eta+2)-\tau(\eta)$ and uses a dynamical, capacity-based approach to bypass Beurling-type estimates. The main contributions are a plane bound $D(\eta)\ge (4-\eta)/2$ for $0\le\eta<2$ and rigorous 2D/3D proofs that recover the optimal known bounds for $\eta=1$ (Kesten for 2D and Lawler for 3D), with results matching or improving existing DLA estimates. The methods rely on capacity increment analysis, Beurling-type estimates in discrete settings, and a discrete Makarov theorem, offering a pathway to continuous settings and broader universality insights for fractal growth phenomena.
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 $\mathbb{Z}^3$.