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Internal DLA and the Gaussian free field

By David Jerison, Lionel Levine and Scott Sheffield


In previous works, we showed that the internal DLA cluster on Z d with t particles is a.s. 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 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

Year: 2010
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