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A Survey on the Stability of (Extended) Linear Sand Pile Model

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Automata and Complexity

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 42))

Abstract

We give a survey of our works on the natural extensions of the well-known Sand Pile Model. These extensions consist of adding outside grains on random columns, allowing sand grains to move from left to right and from right to left, considering cycle graphs and the extension to infinity. We study the reachable configurations and fixed points of each model and show how to compute the set of fixed points, the time of convergence and the distribution of fixed points.

dedicated to the 70th Anniversary of Eric Goles.

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References

  1. Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: an explanation of 1/f noise. Phys Rev Lett 59:381–384

    Article  Google Scholar 

  2. Bj\(\ddot{o}\)rner A,  Lovász L (1992) Chip firing games on directed graphs. J Algebraic Combin 1:305–328

    Google Scholar 

  3. Bj\(\ddot{o}\)rner A, Lovász L, Shor W (1991) Chip-firing games on graphs. Eur J Combin 12:283–291

    Google Scholar 

  4. Brylawski T (1973) The lattice of interger partitions. Discret Math 6:201–219

    Article  Google Scholar 

  5. Cori R, Phan THD, Tran TTH (2013) Signed chip firing games and symmetric sandpile models on the cycles. RAIRO Inf Théor Appl 47(2):133–146

    Article  MathSciNet  Google Scholar 

  6. Cori R, Rossin D (2000) On the sandpile group of dual graphs. Eur J Combin 21:447–459

    Article  MathSciNet  Google Scholar 

  7. Desel J, Kindler E, Vesper T, Walter R (1995) A simplified proof for the self-stabilizing protocol: a game of cards. Inf Proc Lett 54:327–328

    Article  Google Scholar 

  8. Dhar D (1990) Self-organized critical state of sandpile automaton models. Phys Rev Lett 64:1613–1616

    Article  MathSciNet  Google Scholar 

  9. Duchi E, Mantaci R, Phan THD, Rossin D (2006) Bidimensional sand pile and ice pile models. Pure Math Appl (PU.M.A.) 17(1-2):71–96

    Google Scholar 

  10. Durand-Lose J (1996) Grain sorting in the one dimensional sand pile model. Complex Syst 10(3):195–206

    MathSciNet  MATH  Google Scholar 

  11. Durand-Lose J (1998) Parallel transient time of one-dimensional sand pile. Theor Comput Sci 205(1–2):183–193

    Article  MathSciNet  Google Scholar 

  12. Formenti E, Masson B, Pisokas T (2007) Advances in symmetric sandpiles. Fundam Inf 76(1–2):91–112

    MathSciNet  MATH  Google Scholar 

  13. Formenti E, Perrot K, Rémila E (2014) Computational complexity of the avalanche problem on one dimensional kadanoff sandpiles. In: Proceedings of automata ’2014, (LNCS), vol 8996, pp 21–30

    Google Scholar 

  14. Formenti E, Pham TV,  Duong TH, Phan THD, Tran TTH (2014) Fixed-point forms of the parallel symmetric sandpile model. Theor Comput Sci 533:1–14

    Google Scholar 

  15. Goles E, Kiwi MA (1993) Games on line graphs and sand piles. Theor Comput Sci 115:321–349

    Article  MathSciNet  Google Scholar 

  16. Goles E, Morvan M, Phan HD (2002) Lattice structure and convergence of a game of cards. Ann. Combin 6:327–335

    Article  MathSciNet  Google Scholar 

  17. Goles E, Morvan M, Phan HD (2002) Sandpiles and order structure of integer partitions. Discret Appl Math 117:51–64

    Article  MathSciNet  Google Scholar 

  18. Goles E, Morvan M, Phan HD (2002) The structure of linear chip firing game and related models. Theor Comput Sci 270:827–841

    Article  MathSciNet  Google Scholar 

  19. Greene C, Kleiman DJ (1986) Longest chains in the lattice of integer partitions ordered by majorization. Eur J Combin 7:1–10

    Article  MathSciNet  Google Scholar 

  20. Huang S-T (1993) Leader election in uniform rings. ACM Trans Program Lang Syst 15(3):563–573

    Article  Google Scholar 

  21. Kadanoff LP, Nagel SR, Wu L,  Zhou SM (1989) Scaling and universality in avalanches. Phys Rev A 39(12):6524–6537

    Google Scholar 

  22. Karmakar R, Manna SS (2005) Particle hole symmetry in a sandpile model. J Stat Mech: Theory and Exp 2005(01):L01002

    Article  Google Scholar 

  23. Latapy M, Mataci R, Morvan M, Phan HD (2001) Structure of some sand piles model. Theor Comput Sci 262:525–556

    Article  MathSciNet  Google Scholar 

  24. Latapy M, Phan THD (2009) The lattice of integer partitions and its infinite extension. Discret Math 309(6):1357–1367

    Article  MathSciNet  Google Scholar 

  25. Le MH, Phan THD (2009) Integer partitions in discrete dynamical models and ECO method. Vietnam J Math 37(2–3):273–293

    MathSciNet  MATH  Google Scholar 

  26. Le MH, Phan THD Strict partitions and discrete dynamical systems. Theor Comput Sci

    Google Scholar 

  27. Perrot K, Pham TV, Phan THD (2012) On the set of fixed points of the parallel symmetric sand pile model. In: Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems, Discrete Mathematics & Theoretical Computer Science, pp 17–28

    Google Scholar 

  28. Perrot K, Rémila E (2013) Kadanoff sand pile model. avalanche structure and wave shape. Theor Comput Sci 504:52–72

    Google Scholar 

  29. Phan THD (2008) Two sided sand piles model and unimodal sequences. RAIRO Inf Théor Appl 42(3):631–646

    Article  MathSciNet  Google Scholar 

  30. Phan THD, Tran TTH (2010) On the stability of sand piles model. Theor Comput Sci 411(3):594–601

    Article  MathSciNet  Google Scholar 

  31. Spencer J (1986) Balancing vectors in the max norm. Combinatorica 6:55–65

    Article  MathSciNet  Google Scholar 

  32. Stanley RP (1999) Enumerative combinatorics, vol 2. Cambridge University Press, Cambridge

    Book  Google Scholar 

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Acknowledgements

This work was supported by the Vietnam National Foundation for Science ans Technology Development under the grant number NAFOSTED 101.99-2016.16 and by the Vietnam Institute for Advanced Study in Mathematics.

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Authors and Affiliations

  1. Institute of Mathematics - Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam

    Thi Ha Duong Phan

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  1. Thi Ha Duong Phan
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Correspondence to Thi Ha Duong Phan .

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  1. Unconventional Computing Centre, University of the West of England, Bristol, UK

    Andrew Adamatzky

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Phan, T.H.D. (2022). A Survey on the Stability of (Extended) Linear Sand Pile Model. In: Adamatzky, A. (eds) Automata and Complexity. Emergence, Complexity and Computation, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-030-92551-2_16

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  • DOI: https://doi.org/10.1007/978-3-030-92551-2_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-92550-5

  • Online ISBN: 978-3-030-92551-2

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