Scaling Limits for Internal Aggregation Models with Multiple Sources

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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.