Scaling limits for internal aggregation models with multiple sources
 Lionel Levine,
 Yuval Peres
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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 rotorrouter 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 DiaconisFulton smash sum of domains.
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 Title
 Scaling limits for internal aggregation models with multiple sources
 Journal

Journal d'Analyse Mathématique
Volume 111, Issue 1 , pp 151219
 Cover Date
 20100501
 DOI
 10.1007/s1185401000152
 Print ISSN
 00217670
 Online ISSN
 15658538
 Publisher
 The Hebrew University Magnes Press
 Additional Links
 Topics
 Authors

 Lionel Levine ^{(1)} ^{(2)}
 Yuval Peres ^{(3)} ^{(4)}
 Author Affiliations

 1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA
 2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
 3. Departments of Statistics, University of California, Berkeley, CA, 94720, USA
 4. Theory Group, Microsoft Research, Redmond, WA, 98052, USA