Scaling Limits for Internal Aggregation Models with Multiple Sources

Details Scaling Limits for Internal Aggregation Models with Multiple Sources Scaling Limits for Internal Aggregation Models with Multiple Sources

2007-12-20

arxiv.org/abs/0712.3378v2

JOURNAL D'ANALYSE MATH\'EMATIQUE Volume 111, Number 1, 151--219, 2010

Mathematics

Probability

74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in v2: added "least action principle" (Lemma 3.2); small cor

Scientific paper

10.1007/s11854-010-0015-2

We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.

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