Journal d'Analyse Mathématique

, Volume 111, Issue 1, pp 151–219

Scaling limits for internal aggregation models with multiple sources


DOI: 10.1007/s11854-010-0015-2

Cite this article as:
Levine, L. & Peres, Y. JAMA (2010) 111: 151. doi:10.1007/s11854-010-0015-2


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.

Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Departments of StatisticsUniversity of CaliforniaBerkeleyUSA
  4. 4.Theory GroupMicrosoft ResearchRedmondUSA

Personalised recommendations