Internal DLA and the Gaussian free field

Abstract

In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on ${\mathbb{Z}}^d$ with $t$ particles is almost surely spherical up to a maximal error of $O(logt)$ if $d=2$ and $O\left(\sqrt{logt}\right)$ if $d\ge 3$. This paper addresses average error: in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when $d=2$ and of order $r^{1-d/2}$ (for a radius $r$ cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.