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Scaling limits for internal aggregation models with multiple sources

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Abstract

We study the scaling limits of three different aggregation models on ℤd: 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 ℝd. 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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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Lionel Levine

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Lionel Levine

  3. Departments of Statistics, University of California, Berkeley, CA, 94720, USA

    Yuval Peres

  4. Theory Group, Microsoft Research, Redmond, WA, 98052, USA

    Yuval Peres

Authors
  1. Lionel Levine
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  2. Yuval Peres
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Correspondence to Lionel Levine.

Additional information

Supported by an NSF Graduate Research Fellowship, and NSF grant DMS-0605166.

Partially supported by NSF grant DMS-0605166.

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Levine, L., Peres, Y. Scaling limits for internal aggregation models with multiple sources. JAMA 111, 151–219 (2010). https://doi.org/10.1007/s11854-010-0015-2

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  • DOI: https://doi.org/10.1007/s11854-010-0015-2

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