Discrete Balayage and Boundary Sandpile

  • Hayk Aleksanyan
  • Henrik ShahgholianEmail author


We introduce a new lattice growth model, which we call the boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on ℤd (d ≥ 2) onto the boundary of an (a priori) unknown domain. The latter evolves through sandpile dynamics, and has the property that the mass on the boundary is forced to stay below a prescribed threshold. Since finding the domain is part of the problem, the redistribution process is a discrete model of a free boundary problem, whose continuum limit is yet to be understood. We prove general results concerning our model. These include canonical representation of the model in terms of the smallest super-solution among a certain class of functions, uniform Lipschitz regularity of the scaled odometer function, and hence the convergence of a subsequence of the odometer and the visited sites, discrete symmetry properties, as well as directional monotonicity of the odometer function. The latter (in part) implies the Lipschitz regularity of the free boundary of the sandpile.

As a direct application of some of the methods developed in this paper, combined with earlier results on the classical abelian sandpile, we show that the boundary of the scaling limit of an abelian sandpile is locally a Lipschitz graph.


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  1. [1]
    G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), 863–902.MathSciNetzbMATHGoogle Scholar
  2. [2]
    H. Aleksanyan and H. Shahgholian, Perturbed divisible sandpiles and quadrature surfaces, Potential Anal. (2018), Google Scholar
  3. [3]
    A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315.MathSciNetCrossRefGoogle Scholar
  4. [4]
    H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144.MathSciNetzbMATHGoogle Scholar
  5. [5]
    A. Asselah and A. Gaudillière, From logarithmic to subdiffusive polynomial fluctuations for internal DLA and related growth models, Ann. Probab. 41(3A) (2013), 1115–1159.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    P. Bak, C. Tang and K. Wiesenfeld, Self-organized criticality: An explanation of the 1/ f noise, Phys. Rev. A (3) 38 (1988), 364–374.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. Björner, L. Lovász and P. Shor, Chip-firing games on graphs, European J. Combin. 4 (1991), 283–291.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    P. Ebenfelt, B. Gustafsson, D. Khavinson and M. Putinar, Quadrature Domains and Their Applications, Birkhäuser, Basel, 2005.CrossRefzbMATHGoogle Scholar
  9. [9]
    A. Fey, L. Levine and Y. Peres, Growth rates and explosions in sandpiles, J. Stat. Phys. 138 (2010), 143–159.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. Fey and F. Redig, Limiting shapes for deterministic centrally seeded growth models, J. Statist. Phys. 130 (2008), 579–597.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A. Fey and H. Liu, Limiting shapes for a nonabelian sandpile growth model and related cellular automata, J. Cell. Autom. 6 (2011), 353–383.MathSciNetzbMATHGoogle Scholar
  12. [12]
    S. Frómeta and M. Jara, Scaling limit for a long-range divisible sandpile, SIAM J. Math. Anal. 50 (2018), 2317–2361.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Y. Fukai and K. Uchiyama, Potential kernel for two-dimensional random walk, Ann. Probab. 24 (1996), 1979–1992.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Gravner and J. Quastel, Internal DLA and the Stefan problem, Ann. Probab. 28 (2000), 1528–1562.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    B. Gustafsson, Direct and inverse balayage-some newdevelopments in classical potential theory, Nonlinear Anal. 30 (1997), 2557–2565.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    B. Gustafsson and H. Shahgholian, Existence and geometric properties of solutions of a free boundary problem in potential theory, J. Reine Angew. Math. 473 (1996), 137–179.MathSciNetzbMATHGoogle Scholar
  17. [17]
    J. Heinonen, Lectures on Lipschitz Analysis, University of Jyväskylä, Jyväskylä, 2005.zbMATHGoogle Scholar
  18. [18]
    D. Jerison, L. Levine and S. Sheffield, Logarithmic fluctuations for internal DLA, J. Amer. Math. Soc. 25 (2012), 271–301.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    G. Lawler, M. Bramson and D. Griffeath, Internal diffusion limited aggregation, Ann. Probab. 20 (1992), 2117–2140.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    G. Lawler, Subdiffusive fluctuations for internal diffusion limited aggregation, Ann. Probab. 23 (1995), 71–86.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    G. Lawler, Intersections of Random Walks, Birkhäuser, Basel, 1996.zbMATHGoogle Scholar
  22. [22]
    G. Lawler and V. Limic, Random Walk: A Modern Introduction, Cambridge University Press, Cambridge, 2010.CrossRefzbMATHGoogle Scholar
  23. [23]
    L. Levine, Limit Theorems for Internal Aggregation Models, PhD thesis, University of California Berkley, Berkley, CA, 2007.Google Scholar
  24. [24]
    L. Levine, W. Pegden and C. K. Smart, Apollonian structure in the abelian sandpile, Geom. Funct. Anal. 26 (2016), 306–336.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    L. Levine and Y. Peres, Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal. 30 (2009), 1–27.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources, J. Anal. Math. 111 (2010), 151–219.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    C. Lucas, The limiting shape for drifted internal diffusion limited aggregation is a true heat ball, Probab. Theory Relat. Fields 159 (2014), 197–235.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    W. Pegden and C. K. Smart, Convergence of the abelian sandpile, Duke Math. J. 162 (2013), 627–642.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    J. Serrin, A symmetry problem in potential theory, Arch. Ration.Mech. Anal. 43 (1971), 304–318.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    K. Uchiyama, Green’s functions for random walks on ZN, Proc. Lond. Math. Soc. 77 (1998), 215–240.CrossRefGoogle Scholar
  31. [31]
    D. Zidarov, Inverse Gravimetric Problem in Geoprospecting and Geodesy (Developments in Solid Earth Geophysics), Elsevier Science, Amsterdam, 1990.zbMATHGoogle Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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