Advertisement

Potential Analysis

, 30:1 | Cite as

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

  • Lionel Levine
  • Yuval Peres
Open Access
Article

Abstract

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work (Levine and Peres, Indiana Univ Math J 57(1):431–450, 2008). For the shape consisting of \(n=\omega_d r^d\) sites, where ω d is the volume of the unit ball in \(\mathbb{R}^d\), we show that the inradius of the set of occupied sites is at least r − O(logr), while the outradius is at most r + O(r α ) for any α > 1 − 1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n = πr 2 particles, we show that the inradius is at least \(r/\sqrt{3}\), and the outradius is at most \((r+o(r))/\sqrt{2}\). This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions, improving on bounds of Fey and Redig.

Keywords

Abelian sandpile Asymptotic shape Discrete Laplacian Divisible sandpile Growth model Internal diffusion limited aggregation Rotor-router model 

Mathematics Subject Classifications (2000)

Primary 60G50 Secondary 35R35 

References

  1. 1.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cook, M.: The tent metaphor. http://paradise.caltech.edu/~cook/Warehouse/ForPropp/ (2002)
  3. 3.
    Cooper, J.N., Spencer, J.: Simulating a random walk with constant error. Combin. Probab. Comput. 15, 815–822 (2006). http://www.arxiv.org/abs/math.CO/0402323 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Diaconis, P., Fulton, W.: A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec. Torino 49(1), 95–119 (1991)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Fey, A., Redig, F.: Limiting shapes for deterministic centrally seeded growth models. J. Statist. Phys. 130(3), 579–597 (2008). http://arxiv.org/abs/math.PR/0702450 zbMATHCrossRefGoogle Scholar
  6. 6.
    Fukai, Y., Uchiyama, K.: Potential kernel for two-dimensional random walk. Ann. Probab. 24(4), 1979–1992 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kleber, M.: Goldbug variations. Math. Intelligencer 27(1), 55–63 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Koosis, P.: La plus petite majorante surharmonique et son rapport avec l’existence des fonctions entières de type exponentiel jouant le rôle de multiplicateurs. Ann. Inst. Fourier (Grenoble) 33 fasc. (1), 67–107 (1983)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Lawler, G.: Intersections of Random Walks. Birkhäuser, Boston (1996)zbMATHGoogle Scholar
  10. 10.
    Lawler, G., Bramson, M., Griffeath, D.: Internal diffusion limited aggregation. Ann. Probab. 20(4), 2117–2140 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lawler, G.: Subdiffusive fluctuations for internal diffusion limited aggregation. Ann. Probab. 23(1), 71–86 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Le Borgne, Y., Rossin, D.: On the identity of the sandpile group. Discrete Math. 256, 775–790 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Levine, L.: The rotor-router model. Harvard University senior thesis. http://arxiv.org/abs/math/0409407 (2002)
  14. 14.
    Levine, L., Peres, Y.: Spherical asymptotics for the rotor-router model in \(\mathbb{Z}^d\). Indiana Univ. Math. J. 57(1), 431–450 (2008). http://arxiv.org/abs/math/0503251 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Levine, L., Peres, Y.: The rotor-router shape is spherical. Math. Intelligencer 27(3), 9–11 (2005)zbMATHCrossRefGoogle Scholar
  16. 16.
    Levine, L., Peres, Y.: Scaling limits for internal aggregation models with multiple sources. http://arxiv.org/abs/0712.3378 (2007)
  17. 17.
    Lindvall, T.: Lectures on the Coupling Method. Wiley, New York (1992)zbMATHGoogle Scholar
  18. 18.
    Priezzhev, V.B., Dhar, D., Dhar, A., Krishnamurthy, S.: Eulerian walkers as a model of self-organised criticality. Phys. Rev. Lett. 77, 5079–82 (1996)CrossRefGoogle Scholar
  19. 19.
    Propp, J.: Three lectures on quasirandomness. http://faculty.uml.edu/jpropp/berkeley.html (2004)
  20. 20.
    Spitzer, F.: Principles of Random Walk. Springer, New York (1976)zbMATHGoogle Scholar
  21. 21.
    Uchiyama, K.: Green’s functions for random walks on \(\mathbb{Z}^N\). Proc. London Math. Soc. 77(1), 215–240 (1998)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Van den Heuvel, J.: Algorithmic aspects of a chip-firing game. Combin. Probab. Comput. 10(6), 505–529 (2001)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA
  2. 2.Microsoft ResearchRedmondUSA

Personalised recommendations