# Sublogarithmic fluctuations for internal DLA - Mathematics > Probability

Sublogarithmic fluctuations for internal DLA - Mathematics > Probability - Download this document for free, or read online. Document in PDF available to download.

Abstract: We consider internal diffusion limited aggregation in dimension larger thanor equal to two. This is a random cluster growth model, where random walksstart at the origin of the d-dimensional lattice, one at a time, and stopmoving when reaching a site that is not occupied by previous walks. It is knownthat the asymptotic shape of the cluster is a sphere. When the dimension is twoor more, we have shown in a previous paper that the inner resp., outerfluctuations of its radius is at most of order $\log\mathrm{radius}$ resp.,$\log^2\mathrm{radius}$. Using the same approach, we improve the upper boundon the inner fluctuation to $\sqrt{\log\mathrm{radius}}$ when d is largerthan or equal to three. The inner fluctuation is then used to obtain a similarupper bound on the outer fluctuation.

Author: ** Amine Asselah, Alexandre Gaudillière**

Source: https://arxiv.org/