Diffusion-limited aggregation on a tree
- Cite this article as:
- Barlow, M., Pemantle, R. & Perkins, E. Probab Theory Relat Fields (1997) 107: 1. doi:10.1007/s004400050076
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We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to α−n, where α < 1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when α≧ 1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem in first-passage percolation, and as a version of diffusion-limited aggregation (DLA), adjusted so that “fingering” occurs.