Diffusion-Limited Aggregation on Curved Surfaces

TL;DR

This paper develops a theory for diffusion-limited aggregation on curved surfaces using conformal maps, showing that curvature influences aggregate shape but not fractal dimension, with simulations on spheres and pseudo-spheres.

Contribution

It introduces a general framework for DLA on curved surfaces and demonstrates that fractal dimension remains invariant under curvature, supported by simulations on various geometries.

Findings

Fractal dimension is insensitive to curvature when particle size is small.

Curvature affects the global morphology of aggregates.

Multifractal dimensions vary with curvature, increasing from hyperbolic to elliptic geometries.

Abstract

We develop a general theory of transport-limited aggregation phenomena occurring on curved surfaces, based on stochastic iterated conformal maps and conformal projections to the complex plane. To illustrate the theory, we use stereographic projections to simulate diffusion-limited-aggregation (DLA) on surfaces of constant Gaussian curvature, including the sphere ($K>0$) and pseudo-sphere ($K<0$), which approximate "bumps" and "saddles" in smooth surfaces, respectively. Although curvature affects the global morphology of the aggregates, the fractal dimension (in the curved metric) is remarkably insensitive to curvature, as long as the particle size is much smaller than the radius of curvature. We conjecture that all aggregates grown by conformally invariant transport on curved surfaces have the same fractal dimension as DLA in the plane. Our simulations suggest, however, that the multifractal dimensions increase from hyperbolic ($K<0$) to elliptic ($K>0$) geometry, which we attribute to curvature-dependent screening of tip branching.