The dimension of Diffusion Limited Aggregates grown on a line

TL;DR

This paper establishes an exact fractal dimension of 1.5 for off-lattice diffusion limited aggregation grown on a line, using conformal maps to rigorously analyze its self-affinity and scaling properties.

Contribution

It provides the first exact value for the fractal dimension of DLA on a line, advancing the mathematical understanding of this classical growth model.

Findings

Fractal dimension of DLA on a line is exactly 3/2.

Conformal maps enable rigorous proof of self-affinity and scaling.

Mathematical proofs are detailed in arXiv:2008.05792.

Abstract

Diffusion Limited Aggregation (DLA) has served for forty years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references no exact result for the fractal dimension $D$ of DLA is known. In this Letter we announce an exact result for off-lattice DLA grown on a line, $D=3/2$. The result relies on representing DLA with iterated conformal maps, allowing one to prove self-affinity, a proper scaling limit and a well defined fractal dimension. Mathematical proofs of the main results are available in arXiv:2008.05792.