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
Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: an explanation of 1/f noise. Phys Rev Lett 59:381–384
Bj\(\ddot{o}\)rner A,  Lovász L (1992) Chip firing games on directed graphs. J Algebraic Combin 1:305–328
Bj\(\ddot{o}\)rner A, Lovász L, Shor W (1991) Chip-firing games on graphs. Eur J Combin 12:283–291
Brylawski T (1973) The lattice of interger partitions. Discret Math 6:201–219
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
Cori R, Rossin D (2000) On the sandpile group of dual graphs. Eur J Combin 21:447–459
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
Dhar D (1990) Self-organized critical state of sandpile automaton models. Phys Rev Lett 64:1613–1616
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
Durand-Lose J (1996) Grain sorting in the one dimensional sand pile model. Complex Syst 10(3):195–206
Durand-Lose J (1998) Parallel transient time of one-dimensional sand pile. Theor Comput Sci 205(1–2):183–193
Formenti E, Masson B, Pisokas T (2007) Advances in symmetric sandpiles. Fundam Inf 76(1–2):91–112
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
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
Goles E, Kiwi MA (1993) Games on line graphs and sand piles. Theor Comput Sci 115:321–349
Goles E, Morvan M, Phan HD (2002) Lattice structure and convergence of a game of cards. Ann. Combin 6:327–335
Goles E, Morvan M, Phan HD (2002) Sandpiles and order structure of integer partitions. Discret Appl Math 117:51–64
Goles E, Morvan M, Phan HD (2002) The structure of linear chip firing game and related models. Theor Comput Sci 270:827–841
Greene C, Kleiman DJ (1986) Longest chains in the lattice of integer partitions ordered by majorization. Eur J Combin 7:1–10
Huang S-T (1993) Leader election in uniform rings. ACM Trans Program Lang Syst 15(3):563–573
Kadanoff LP, Nagel SR, Wu L,  Zhou SM (1989) Scaling and universality in avalanches. Phys Rev A 39(12):6524–6537
Karmakar R, Manna SS (2005) Particle hole symmetry in a sandpile model. J Stat Mech: Theory and Exp 2005(01):L01002
Latapy M, Mataci R, Morvan M, Phan HD (2001) Structure of some sand piles model. Theor Comput Sci 262:525–556
Latapy M, Phan THD (2009) The lattice of integer partitions and its infinite extension. Discret Math 309(6):1357–1367
Le MH, Phan THD (2009) Integer partitions in discrete dynamical models and ECO method. Vietnam J Math 37(2–3):273–293
Le MH, Phan THD Strict partitions and discrete dynamical systems. Theor Comput Sci
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
Perrot K, Rémila E (2013) Kadanoff sand pile model. avalanche structure and wave shape. Theor Comput Sci 504:52–72
Phan THD (2008) Two sided sand piles model and unimodal sequences. RAIRO Inf Théor Appl 42(3):631–646
Phan THD, Tran TTH (2010) On the stability of sand piles model. Theor Comput Sci 411(3):594–601
Spencer J (1986) Balancing vectors in the max norm. Combinatorica 6:55–65
Stanley RP (1999) Enumerative combinatorics, vol 2. Cambridge University Press, Cambridge
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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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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