Internal DLA and the Gaussian free field

In previous works, we showed that the internal DLA cluster on Z with t particles is almost surely spherical up to a maximal error of O(log t) if d = 2 and O( √ log t) 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. ∗Partially supported by NSF grant DMS-1069225. †Supported by an NSF Postdoctoral Research Fellowship and NSF grant DMS-1105960. ‡Partially supported by NSF grant DMS-0645585.