Mathematics > Probability
[Submitted on 20 Dec 2007 (v1), last revised 2 May 2009 (this version, v2)]
Title:Scaling Limits for Internal Aggregation Models with Multiple Sources
View PDFAbstract: 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.
Submission history
From: Lionel Levine [view email][v1] Thu, 20 Dec 2007 11:23:23 UTC (775 KB)
[v2] Sat, 2 May 2009 04:50:53 UTC (1,133 KB)
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