Sub-logarithmic fluctuations for internal DLA

Details Sub-logarithmic fluctuations for internal DLA Sub-logarithmic fluctuations for internal DLA

2010-11-20

arxiv.org/abs/1011.4592v3

Mathematics

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19 pages, final version

Scientific paper

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.

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