Internal DLA and the Gaussian free field

In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on Zd with t particles is almost surely spherical up to a maximal error of O(logt) if d=2 and O(logt) if d≥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 r1−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.