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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
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.
H. Aleksanyan and H. Shahgholian, Perturbed divisible sandpiles and quadrature surfaces, Potential Anal. (2018), https://doi.org/10.1007/s11118-018-9722-6
A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315.
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.
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.
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.
A. Björner, L. Lovász and P. Shor, Chip-firing games on graphs, European J. Combin. 4 (1991), 283–291.
P. Ebenfelt, B. Gustafsson, D. Khavinson and M. Putinar, Quadrature Domains and Their Applications, Birkhäuser, Basel, 2005.
A. Fey, L. Levine and Y. Peres, Growth rates and explosions in sandpiles, J. Stat. Phys. 138 (2010), 143–159.
A. Fey and F. Redig, Limiting shapes for deterministic centrally seeded growth models, J. Statist. Phys. 130 (2008), 579–597.
A. Fey and H. Liu, Limiting shapes for a nonabelian sandpile growth model and related cellular automata, J. Cell. Autom. 6 (2011), 353–383.
S. Frómeta and M. Jara, Scaling limit for a long-range divisible sandpile, SIAM J. Math. Anal. 50 (2018), 2317–2361.
Y. Fukai and K. Uchiyama, Potential kernel for two-dimensional random walk, Ann. Probab. 24 (1996), 1979–1992.
J. Gravner and J. Quastel, Internal DLA and the Stefan problem, Ann. Probab. 28 (2000), 1528–1562.
B. Gustafsson, Direct and inverse balayage-some newdevelopments in classical potential theory, Nonlinear Anal. 30 (1997), 2557–2565.
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.
J. Heinonen, Lectures on Lipschitz Analysis, University of Jyväskylä, Jyväskylä, 2005.
D. Jerison, L. Levine and S. Sheffield, Logarithmic fluctuations for internal DLA, J. Amer. Math. Soc. 25 (2012), 271–301.
G. Lawler, M. Bramson and D. Griffeath, Internal diffusion limited aggregation, Ann. Probab. 20 (1992), 2117–2140.
G. Lawler, Subdiffusive fluctuations for internal diffusion limited aggregation, Ann. Probab. 23 (1995), 71–86.
G. Lawler, Intersections of Random Walks, Birkhäuser, Basel, 1996.
G. Lawler and V. Limic, Random Walk: A Modern Introduction, Cambridge University Press, Cambridge, 2010.
L. Levine, Limit Theorems for Internal Aggregation Models, PhD thesis, University of California Berkley, Berkley, CA, 2007.
L. Levine, W. Pegden and C. K. Smart, Apollonian structure in the abelian sandpile, Geom. Funct. Anal. 26 (2016), 306–336.
L. Levine and Y. Peres, Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal. 30 (2009), 1–27.
L. Levine and Y. Peres, Scaling limits for internal aggregation models with multiple sources, J. Anal. Math. 111 (2010), 151–219.
C. Lucas, The limiting shape for drifted internal diffusion limited aggregation is a true heat ball, Probab. Theory Relat. Fields 159 (2014), 197–235.
W. Pegden and C. K. Smart, Convergence of the abelian sandpile, Duke Math. J. 162 (2013), 627–642.
J. Serrin, A symmetry problem in potential theory, Arch. Ration.Mech. Anal. 43 (1971), 304–318.
K. Uchiyama, Green’s functions for random walks on ZN, Proc. Lond. Math. Soc. 77 (1998), 215–240.
D. Zidarov, Inverse Gravimetric Problem in Geoprospecting and Geodesy (Developments in Solid Earth Geophysics), Elsevier Science, Amsterdam, 1990.
H. A. was supported by a postdoctoral fellowship from the Knut and AliceWallenberg Foundation.
H. Sh. was partially supported by the Swedish Research Council.
About this article
Cite this article
Aleksanyan, H., Shahgholian, H. Discrete Balayage and Boundary Sandpile. JAMA 138, 361–403 (2019). https://doi.org/10.1007/s11854-019-0037-3