Diamond Aggregation

Details Diamond Aggregation Diamond Aggregation

2009-05-09

arxiv.org/abs/0905.1361v2

Math. Proc. Cambridge Phil. Soc. 149(2): 351-372 (2010)

Mathematics

Probability

v2 addresses referee comments, new section on the abelian property

Scientific paper

10.1017/S030500411000006X

Internal diffusion-limited aggregation is a growth model based on random walk in Z^d. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in Z^2 for which the limiting shape is a diamond. Certain of these walks -- those with a directional bias toward the origin -- have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

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