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May 2013 Sublogarithmic fluctuations for internal DLA
Amine Asselah, Alexandre Gaudillière
Ann. Probab. 41(3A): 1160-1179 (May 2013). DOI: 10.1214/11-AOP735


We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where random walks start at the origin of the $d$-dimensional lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. It is known that the asymptotic shape of the cluster is a sphere. When the dimension is two or more, we have shown in a previous paper that the inner (resp., outer) fluctuations of its radius is at most of order $\log(\mathrm{radius})$ [resp., $\log^{2}(\mathrm{radius})$]. Using the same approach, we improve the upper bound on the inner fluctuation to $\sqrt{\log(\mathrm{radius})}$ when $d$ is larger than or equal to three. The inner fluctuation is then used to obtain a similar upper bound on the outer fluctuation.


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Amine Asselah. Alexandre Gaudillière. "Sublogarithmic fluctuations for internal DLA." Ann. Probab. 41 (3A) 1160 - 1179, May 2013.


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

zbMATH: 1274.60286
MathSciNet: MR3098674
Digital Object Identifier: 10.1214/11-AOP735

Primary: 60J45 , 60K35 , 82B24

Keywords: Cluster growth , internal diffusion limited aggregation , logarithmic fluctuations , Random walk , shape theorem

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 3A • May 2013
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