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Diffusion-Limited Aggregation on Curved Surfaces
Statistical Mechanics · Submitted 15 Mar 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.91 | 0.94 | 0.06 | asserted 1.00 |
| s1 | 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. | method 0.93 | 0.94 | 0.05 | asserted 0.99 |
| s2 | 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. | result 1.00 | 0.87 | 0.53 | asserted 1.00 |
| s3 | We conjecture that all aggregates grown by conformally invariant transport on curved surfaces have the same fractal dimension as DLA in the plane. | interpretation 0.81 | 0.19 | 0.42 | speculative 1.00 |
| s4 | 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. | result 0.72 | 0.95 | 0.12 | hedged possibility 0.99 |
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Original 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.
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