The Annals of Probability

Sublogarithmic fluctuations for internal DLA

Amine Asselah and Alexandre Gaudillière

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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.

Article information

Ann. Probab., Volume 41, Number 3A (2013), 1160-1179.

First available in Project Euclid: 29 April 2013

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B24: Interface problems; diffusion-limited aggregation 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Internal diffusion limited aggregation cluster growth random walk shape theorem logarithmic fluctuations


Asselah, Amine; Gaudillière, Alexandre. Sublogarithmic fluctuations for internal DLA. Ann. Probab. 41 (2013), no. 3A, 1160--1179. doi:10.1214/11-AOP735.

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