There are many real-world processes exhibiting fractal growing shapes – from mineral deposition and coral growth to lightning strikes, and in many of them growth is related to diffusion properties. We will discuss two seminal models: Diffusion Limited Aggregation was introduced by Witten and Sanders in 1981 and was generalized to Dielectric Breakdown Model by Niemayer et al shortly afterwards. Numerically they approximate very well a wide range of physical phenomena.
However, despite a very simple definition (DLA cluster grows by attaching particles undergoing Brownian motion when they hit the aggregate), very little is understood today, and even less is known rigorously – essentially, only the famous Harry Kesten upper bound on the DLA growth.
We will try to show the flavor of these models and present some new results. Based on joint work with Ilya Losev.