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Pré-Publication, Document De Travail Année : 2011

Sub-logarithmic fluctuations for internal DLA

Résumé

We consider internal diffusion limited aggregation in dimension larger 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 not occupied by previous walks. It is known that the asymptotic shape of the cluster is a sphere. When dimension is two or more, we have shown that the inner (resp. outer) fluctuations of its radius is at most of order $\log(\text{radius})$ (resp. $\log^2(\text{radius})$). Using the same approach, we improve the upper bound on the inner fluctuation to $\sqrt{\log(\text{radius})}$ 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.

Dates et versions

hal-00795849 , version 1 (01-03-2013)

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Amine Asselah, Alexandre Gaudilliere. Sub-logarithmic fluctuations for internal DLA. 2011. ⟨hal-00795849⟩
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