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Diamond aggregation

Published online by Cambridge University Press:  10 May 2010

WOUTER KAGER  and
LIONEL LEVINE
WOUTER KAGER
Affiliation:
Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. http://www.few.vu.nl/~wkager e-mail: [email protected]
LIONEL LEVINE
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.http://math.mit.edu/~levine e-mail: [email protected]
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Abstract

Internal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 351 - 372
Copyright
Copyright © Cambridge Philosophical Society 2010

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