Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Diamond aggregation

WOUTER KAGERa1 and LIONEL LEVINEa2

a1 Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. http://www.few.vu.nl/~wkager e-mail: wkager@few.vu.nl

a2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. http://math.mit.edu/~levine e-mail: levine@math.mit.edu

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.

(Received June 29 2009)

(Revised November 12 2009)

(Online publication May 10 2010)