Abstract
We consider a variation of the Hastings-Levitov model HL(0) for random growth in which the growing cluster consists of two competing regions. We allow the size of successive particles to depend both on the region in which the particle is attached, and the harmonic measure carried by that region. We identify conditions under which one can ensure coexistence of both regions. In particular, we consider whether it is possible for the process giving the relative harmonic measures of the regions to converge to a non-trivial ergodic limit.
Highlights
We consider planar random growth models in which clusters grow by the successive attachment of single particles
In physical models for random growth, Laplacian random growth models, the growth rate along the cluster boundary depends on the harmonic measure of the cluster boundary
Dependency on harmonic measure is introduced by allowing the growth of each competing region to depend on the relative harmonic measure of that region
Summary
We consider planar random growth models in which clusters grow by the successive attachment of single particles. The simplest model of this type is the HL(0) model, proposed by Hastings and Levitov [2], in which clusters are constructed as successive compositions of i.i.d. mappings. This model has been well studied (see [7, 8] amongst others). In physical models for random growth, Laplacian random growth models, the growth rate along the cluster boundary depends on the harmonic measure of the cluster boundary. This dependency makes the analysis considerably less tractable. We explore whether it is possible for both regions to coexist indefinitely (in the sense that there is a positive probability that each region has positive harmonic measure for all time), or whether it is always the case that one region will dominate to the exclusion of the other
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