Diffusion-limited aggregation on curved surfaces

Abstract

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