Sublogarithmic fluctuations for internal DLA

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(radius) [resp., log²(radius)]. Using the same approach, we improve the upper bound on the inner fluctuation to √log(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.