Probability Theory and Related Fields

, Volume 107, Issue 1, pp 1–60

Diffusion-limited aggregation on a tree

Authors

  • Martin T. Barlow
    • Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada (e-mail: barlow@math.ubc.ca)
  • Robin Pemantle
    • Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, USA
  • Edwin A. Perkins
    • Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada (e-mail: barlow@math.ubc.ca)

DOI: 10.1007/s004400050076

Cite this article as:
Barlow, M., Pemantle, R. & Perkins, E. Probab Theory Relat Fields (1997) 107: 1. doi:10.1007/s004400050076

Summary.

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.

Mathematics Subject Classification (1991): 60K4060K3060K9960F0560F1560K35
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Copyright information

© Springer-Verlag Berlin Heidelberg 1997