Average shape of transport-limited aggregates

We study the relation between stochastic and continuous transport-limited growth models. We derive a nonlinear integro-differential equation for the average shape of stochastic aggregates, whose mean-field approximation is the corresponding continuous equation. Focusing on the advection-diffusion-limited aggregation (ADLA) model, we show that the average shape of the stochastic growth is similar, but not identical, to the corresponding continuous dynamics. Similar results should apply to DLA, thus explaining the known discrepancies between average DLA shapes and viscous fingers in a channel geometry.