Diffusion limited aggregation on a tree

2004/04/05 by MArtin T. Barlow, Robin Pemantle, Edwin A. Perkins
Mathematics · #math.PR #msc:60K40 #msc:60K30 #msc:60F05 #msc:60F15 #msc:60K35

paper · pdf

published as Prob. Th. Rel. Fields, 107, 1 - 60 (1997) · 56 pages

arxiv created 2004/04/05 · arxiv updated 2009/12/01

Abstract

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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