Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Diamond aggregation


a1 Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. e-mail:

a2 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e-mail:


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)