IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v263y1999i1p4-25.html
   My bibliography  Save this article

The Abelian sandpile and related models

Author

Listed:
  • Dhar, Deepak

Abstract

The Abelian sandpile model is the simplest analytically tractable model of self-organized criticality. This paper presents a brief review of known results about the model. The Abelian group structure of the algebra of operators allows an exact calculation of many of its properties. In particular, when there is a preferred direction, one can calculate all the critical exponents characterizing the distribution of avalanche-sizes in all dimensions. For the undirected case, the model is related to q → 0 state Potts model. This enables exact calculation of some exponents in two dimensions, and there are some conjectures about others. We also discuss a generalization of the model to a network of communicating reactive processors. This includes sandpile models with stochastic toppling rules as a special case. We also consider a nondashAbelian stochastic variant, which lies in a different universality class, related to directed percolation.

Suggested Citation

  • Dhar, Deepak, 1999. "The Abelian sandpile and related models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 263(1), pages 4-25.
    • Handle: RePEc:eee:phsmap:v:263:y:1999:i:1:p:4-25
    DOI: 10.1016/S0378-4371(98)00493-2
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437198004932
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/S0378-4371(98)00493-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ivashkevich, E.V. & Priezzhev, V.B., 1998. "Introduction to the sandpile model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 254(1), pages 97-116.
    2. Ivashkevich, E.V. & Ktitarev, D.V. & Priezzhev, V.B., 1994. "Waves of topplings in an Abelian sandpile," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 209(3), pages 347-360.
    3. Daerden, Frank & Vanderzande, Carlo, 1998. "Sandpiles on a Sierpinski gasket," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 256(3), pages 533-546.
    4. Ananda Rakshit & Nirup Krishnamurthy & Gang Yu, 1996. "System Operations Advisor: A Real-Time Decision Support System for Managing Airline Operations at United Airlines," Interfaces, INFORMS, vol. 26(2), pages 50-58, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hay, M.J. & Schiff, J. & Fisch, N.J., 2017. "On extreme points of the diffusion polytope," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 225-236.
    2. Liudmila Zhilyakova, 2021. "Single-Threshold Model Resource Network and Its Double-Threshold Modifications," Mathematics, MDPI, vol. 9(12), pages 1-34, June.
    3. Tadić, Bosiljka, 2001. "Dynamics of directed graphs: the world-wide Web," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 273-284.
    4. Liudmila Zhilyakova & Vasily Koreshkov & Nadezhda Chaplinskaia, 2022. "Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model," Mathematics, MDPI, vol. 10(21), pages 1-18, November.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shapoval, A.B. & Shnirman, M.G., 2012. "The BTW mechanism on a self-similar image of a square: A path to unexpected exponents," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 15-20.
    2. Gang Yu & Michael Argüello & Gao Song & Sandra M. McCowan & Anna White, 2003. "A New Era for Crew Recovery at Continental Airlines," Interfaces, INFORMS, vol. 33(1), pages 5-22, February.
    3. Jay M. Rosenberger & Ellis L. Johnson & George L. Nemhauser, 2003. "Rerouting Aircraft for Airline Recovery," Transportation Science, INFORMS, vol. 37(4), pages 408-421, November.
    4. Bard, Jonathan F. & Mohan, Dinesh Natarajan, 2008. "Reallocating arrival slots during a ground delay program," Transportation Research Part B: Methodological, Elsevier, vol. 42(2), pages 113-134, February.
    5. Brouer, Berit D. & Dirksen, Jakob & Pisinger, David & Plum, Christian E.M. & Vaaben, Bo, 2013. "The Vessel Schedule Recovery Problem (VSRP) – A MIP model for handling disruptions in liner shipping," European Journal of Operational Research, Elsevier, vol. 224(2), pages 362-374.
    6. Obrad Babić & Milica Kalić & Goran Pavković & Slavica Dožić & Mirjana Čangalović, 2010. "Heuristic approach to the airline schedule disturbances problem," Transportation Planning and Technology, Taylor & Francis Journals, vol. 33(3), pages 257-280, February.
    7. Zhang, Yu, 2008. "Real-time Inter-modal Strategies for Airline Schedule Perturbation Recovery and Airport Congestion Mitigation under Collaborative Decision Making (CDM)," University of California Transportation Center, Working Papers qt2k44c9tx, University of California Transportation Center.
    8. M. Ausloos & K. Ivanova & N. Vandewalle, 2001. "Crashes : symptoms, diagnoses and remedies," Papers cond-mat/0104127, arXiv.org, revised Apr 2001.
    9. Thengvall, Benjamin G. & Yu, Gang & Bard, Jonathan F., 2001. "Multiple fleet aircraft schedule recovery following hub closures," Transportation Research Part A: Policy and Practice, Elsevier, vol. 35(4), pages 289-308, May.
    10. T. Andersson * & P. Värbrand, 2004. "The flight perturbation problem," Transportation Planning and Technology, Taylor & Francis Journals, vol. 27(2), pages 91-117, March.
    11. Dennis F.X. Mathaisel, 1997. "Decision Support for Airline Schedule Planning," Journal of Combinatorial Optimization, Springer, vol. 1(3), pages 251-275, October.
    12. Joyce W. Yen & John R. Birge, 2006. "A Stochastic Programming Approach to the Airline Crew Scheduling Problem," Transportation Science, INFORMS, vol. 40(1), pages 3-14, February.
    13. Stojkovic, Goran & Soumis, François & Desrosiers, Jacques & Solomon, Marius M., 2002. "An optimization model for a real-time flight scheduling problem," Transportation Research Part A: Policy and Practice, Elsevier, vol. 36(9), pages 779-788, November.
    14. Kaiser, Robin & Sava-Huss, Ecaterina, 2024. "Random rotor walks and i.i.d. sandpiles on Sierpiński graphs," Statistics & Probability Letters, Elsevier, vol. 209(C).
    15. Sokolov, Andrey & Melatos, Andrew & Kieu, Tien & Webster, Rachel, 2015. "Memory on multiple time-scales in an Abelian sandpile," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 428(C), pages 295-301.
    16. Paul M. Carlson, 2000. "Exploiting the Opportunities of Collaborative Decision Making: A Model and Efficient Solution Algorithm for Airline Use," Transportation Science, INFORMS, vol. 34(4), pages 381-393, November.
    17. Moudani, Walid El & Mora-Camino, Félix, 2000. "A dynamic approach for aircraft assignment and maintenance scheduling by airlines," Journal of Air Transport Management, Elsevier, vol. 6(4), pages 233-237.
    18. Hu, Yuzhen & Song, Yan & Zhao, Kang & Xu, Baoguang, 2016. "Integrated recovery of aircraft and passengers after airline operation disruption based on a GRASP algorithm," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 87(C), pages 97-112.
    19. S Eom & E Kim, 2006. "A survey of decision support system applications (1995–2001)," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(11), pages 1264-1278, November.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:263:y:1999:i:1:p:4-25. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.