We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the attachment point of each successive particle is distributed according to harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability.
@cite_13: We consider a family of growth models defined using conformal maps in which the local growth rate is determined by (| _n'|^ - ), where ( _n ) is the aggregate map for n particles. We establish a scaling limit result in which strong feedback in the growth rule leads to one-dimensional limits in the form of straight slits. More precisely, we exhibit a phase transition in the ancestral structure of the growing clusters: for ( >1 ), aggregating particles attach to their immediate predecessors with high probability, while for ( <1 ) almost surely this does not happen.
In this paper, we approach the question of a phase transition in ALE( @math ) at @math from the opposite side to @cite_13 by focussing on the regime @math and showing the existence of a phase transition at the level of fluctuations. Our results apply in a more universal setting as we can consider a wide class of particle shapes, not just slits.