During the last 50 years, one could and still can can notice a very strong interest in models of random growth of clusters. The first models were often set up on a lattice or Sander’s diffusion limited aggregation (DLA). There were also models related to the family of dielectric breakdown models. All of these models allowed for a description of the asymptotic behaviour of large clusters. Many computational investigations have revealed structures of fractal type, which, in some cases, resemble natural phenomena. These investigations also have shown sensitivity to details of implementation (e.g. different fractal dimensions). Such cases suggested that lattice-based models may not be the most effective way to describe these physical structures. In addition, lattice models are sometimes difficult to analyse.
In 1998, a family of continuum growth models in terms of sequences of iterated conformal maps was formulated. These models are indexed by a parameter \(\alpha\in [0, 2]\) – for \(\alpha = 1\) we have an Eden model whereas for \(\alpha = 2\) there is a DLA model.
This pape deals with the case \(\alpha = 0\) but in the limiting regime where the particle diameter \(\delta\) becomes small and where the size of the cluster is of order 1 or larger. This paper presents a precise description of the macroscopic shape and growth dynamics of clusters; the evolution of the harmonic measure on the cluster boundary converges to the coalescing Brownian flow, also known as the Brownian web.