In previous works, we showed that the internal diffusion-limited aggregation (DLA) cluster on with particles is almost surely spherical up to a maximal error of if and if . This paper addresses average error: in a certain sense, the average deviation of internal DLA from its mean shape is of constant order when and of order (for a radius cluster) in general. Appropriately normalized, the fluctuations (taken over time and space) scale to a variant of the Gaussian free field.
"Internal DLA and the Gaussian free field." Duke Math. J. 163 (2) 267 - 308, 1 February 2014. https://doi.org/10.1215/00127094-2430259