Diffusion limited aggregation on a tree

Details Diffusion limited aggregation on a tree Diffusion limited aggregation on a tree

2004-04-05

arxiv.org/abs/math/0404089v1

Prob. Th. Rel. Fields, 107, 1 - 60 (1997)

Mathematics

Probability

56 pages

Scientific paper

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 alpha^{-n}, where alpha<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 alpha>=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.

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