# 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 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.

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## References

- [1]D. Aharonov and H. S. Shapiro,
*Domains on which analytic functions satisfy quadrature identities*, J. Analyse Math.**30**(1976), 39–73.zbMATHCrossRefMathSciNetGoogle Scholar - [2]N. Alon and J. H. Spencer,
*The Probabilistic Method*, John Wiley & Sons Inc., New York, 1992.zbMATHGoogle Scholar - [3]S. Axler, P. Bourdon and W. Ramey,
*Harmonic Function Theory*, 2nd ed., Springer, Berlin, 2001.zbMATHGoogle Scholar - [4]L. A. Caffarelli,
*The obstacle problem revisited*, J. Fourier Anal. Appl.**4**(1998), 383–402.zbMATHCrossRefMathSciNetGoogle Scholar - [5]L. A. Caffarelli, L. Karp and H. Shahgholian,
*Regularity of a free boundary problem with application to the Pompeiu problem*, Ann. of Math. (2)**151**(2000), 269–292.zbMATHCrossRefMathSciNetGoogle Scholar - [6]D. Crowdy,
*Quadrature domains and fluid dynamics*, in*Quadrature Domains and Their Applications*, Oper. Theory Adv. Appl.**156**(2005), 113–129.CrossRefMathSciNetGoogle Scholar - [7]P. Diaconis and W. Fulton,
*A growth model, a game, an algebra, Lagrange inversion, and characteristic classes*, Rend. Sem. Mat. Univ. Pol. Torino**49**(1991), 95–119.zbMATHMathSciNetGoogle Scholar - [8]J. L. Doob,
*Classical Potential Theory and Its Probabilistic Counterpart*, Springer, Berlin, 1984.zbMATHGoogle Scholar - [9]L. C. Evans,
*Partial Differential Equations*, Amer. Math. Soc., Providence, RI, 1998.zbMATHGoogle Scholar - [10]A. Friedman,
*Variational Principles and Free-Boundary Problems*, John Wiley & Sons Inc., New York, 1982.zbMATHGoogle Scholar - [11]Y. Fukai and K. Uchiyama,
*Potential kernel for two-dimensional random walk*, Ann. Probab.**24**(1996), 1979–1992.zbMATHCrossRefMathSciNetGoogle Scholar - [12]J. Gravner and J. Quastel,
*Internal DLA and the Stefan problem*, Ann. Probab.**28**(2000), 1528–1562.zbMATHCrossRefMathSciNetGoogle Scholar - [13]B. Gustafsson,
*Quadrature Identities and the Schottky double*, Acta Appl. Math.**1**(1983), 209–240.zbMATHCrossRefMathSciNetGoogle Scholar - [14]B. Gustafsson,
*Singular and special points on quadrature domains from an algebraic geometric point of view*, J. Analyse Math.**51**(1988), 91–117.zbMATHCrossRefMathSciNetGoogle Scholar - [15]B. Gustafsson and M. Sakai,
*Properties of some balayage operators with applications to quadrature domains and moving boundary problems*, Nonlinear Anal.**22**(1994), 1221–1245.zbMATHCrossRefMathSciNetGoogle Scholar - [16]B. Gustafsson and H. S. Shapiro,
*What is a quadrature domain?*in*Quadrature Domains and Their Applications*, Birkhäuser, Basel, 2005, pp. 1–25.CrossRefGoogle Scholar - [17]L. Karp and A. S. Margulis,
*Newtonian potential theory for unbounded sources and applications to free boundary problems*, J. Analyse Math.**70**(1996), 1–63.zbMATHCrossRefMathSciNetGoogle Scholar - [18]L. Karp and H. Shahgholian,
*Regularity of a free boundary problem*, J. Geom. Anal.**9**(1999), 653–669.zbMATHMathSciNetGoogle Scholar - [19]G. Lawler,
*Intersections of Random Walks*, Birkhäuser, Basel, 1996.zbMATHGoogle Scholar - [20]G. Lawler, M. Bramson and D. Griffeath,
*Internal diffusion limited aggregation*, Ann. Probab.**20**(1992), 2117–2140.zbMATHCrossRefMathSciNetGoogle Scholar - [21]L. Levine and Y. Peres,
*Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile*, Potential Anal.**30**(2009), 1–27. http://arxiv.org/abs/0704.0688.zbMATHCrossRefMathSciNetGoogle Scholar - [22]E. H. Lieb and M. Loss,
*Analysis*, 2nd ed., Amer. Math. Soc., Providence, RI, 2001.zbMATHGoogle Scholar - [23]T. Lindvall,
*Lectures on the Coupling Method*, John Wiley & Sons Inc., New York, 1992.zbMATHGoogle Scholar - [24]V. B. Priezzhev, D. Dhar, A. Dhar, and S. Krishnamurthy,
*Eulerian walkers as a model of self-organised criticality*, Phys. Rev. Lett.**77**(1996), 5079–5082.CrossRefGoogle Scholar - [25]S. Richardson,
*Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel*, J. Fluid Mech.**56**(1972), 609–618.zbMATHCrossRefGoogle Scholar - [26]
- [27]M. Sakai,
*Solutions to the obstacle problem as Green potentials*, J. Analyse Math.**44**(1984/85), 97–116.CrossRefMathSciNetGoogle Scholar - [28]H. Shahgholian,
*On quadrature domains and the Schwarz potential*, J. Math. Anal. Appl.**171**(1992), 61–78.zbMATHCrossRefMathSciNetGoogle Scholar - [29]H. S. Shapiro,
*The Schwarz Function and its Generalization to Higher Dimensions*, John Wiley & Sons Inc., New York, 1992.zbMATHGoogle Scholar - [30]K. Uchiyama,
*Green’s functions for random walks on*ℤ^{N}, Proc. London Math. Soc.**77**(1998), 215–240.CrossRefMathSciNetGoogle Scholar - [31]A. N. Varchenko and P. I. Etingof,
*Why the Boundary of a Round Drop Becomes a Curve of Order Four*, Amer. Math. Soc., Providence, RI, 1992.zbMATHGoogle Scholar