Scaling Limits for Internal Aggregation Models with Multiple Sources

Abstract

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.

74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in v2: added "least action principle" (Lemma 3.2); small corrections in section 4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version); expanded section 6.1

References in corpus (1)

• Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

Cited by in corpus (29)

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• Discrete low-discrepancy sequences
• A growth model based on the arithmetic $Z$-game
• Convergence of the random Abelian sandpile
• Discrete analogue computing with rotor-routers
• Scaling limit for a long-range divisible sandpile
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• Formation of an interface by competitive erosion
• Asymptotics of the odometer function for the internal Diffusion Limited Aggregation model
• Quasistatic Droplet percolation
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• Partial Balayage and a Generalization of the Divisible Sandpile Model
• On Hele-Shaw problems arising as scaling limits
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• Some geometric relations for equipotential curves
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