Scaling limits for planar aggregation with subcritical fluctuations

Abstract

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 location of each successive particle is distributed according to the density of 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. The methodology developed in this paper provides a blueprint for analysing more general random growth models, such as the Hastings-Levitov family.

arXiv comment62 pages, 7 figures. In version 4, we have moved some of the technical estimates into appendices, to aid readability

References in corpus (7)

• Laplacian growth as one-dimensional turbulence
• Hastings-Levitov aggregation in the small-particle limit
• Scaling limits of anisotropic Hastings-Levitov clusters
• Small particle limits in a regularized Laplacian random growth model
• One-dimensional scaling limits in a planar Laplacian random growth model
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Cited by in corpus (4)

• One-dimensional scaling limits in a planar Laplacian random growth model
• Growth of Stationary Hastings-Levitov
• Random Growth via Gradient Flow Aggregation
• SLE scaling limits for a Laplacian random growth model