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May 2013 From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models
Amine Asselah, Alexandre Gaudillière
Ann. Probab. 41(3A): 1115-1159 (May 2013). DOI: 10.1214/12-AOP762

Abstract

We consider a cluster growth model on ${\mathbb{Z}}^{d}$, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension $d\ge2$. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachère in our Appendix) on the expected time spent by a random walk inside an annulus.

Citation

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Amine Asselah. Alexandre Gaudillière. "From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models." Ann. Probab. 41 (3A) 1115 - 1159, May 2013. https://doi.org/10.1214/12-AOP762

Information

Published: May 2013
First available in Project Euclid: 29 April 2013

zbMATH: 1283.60117
MathSciNet: MR3098673
Digital Object Identifier: 10.1214/12-AOP762

Subjects:
Primary: 60J45, 60K35, 82B24

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3A • May 2013
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