Diamond aggregation
- 10 May 2010
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 149 (2) , 351-372
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.Keywords
All Related Versions
This publication has 14 references indexed in Scilit:
- Reflected Brownian motion in generic triangles and wedgesStochastic Processes and their Applications, 2007
- Internal Diffusion Limited Aggregation on Discrete Groups Having Exponential GrowthProbability Theory and Related Fields, 2006
- Reflected planar Brownian motions, intertwining relations and crossing probabilitiesAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2004
- Agrégation limitée par diffusion interne dans un environement aléatoireAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2003
- Internal DLA and the Stefan problemThe Annals of Probability, 2000
- Subdiffusive Fluctuations for Internal Diffusion Limited AggregationThe Annals of Probability, 1995
- Internal Diffusion Limited AggregationThe Annals of Probability, 1992
- Markov FunctionsThe Annals of Probability, 1981
- On a Relation Between an Estimate of the Remainder in the Central Limit Theorem and the Law of the Iterated LogarithmTheory of Probability & Its Applications, 1966
- Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian lawActa Mathematica, 1945