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The Abelian sandpile and related models

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  • 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.

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  • 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
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    1. 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.
    2. Daerden, Frank & Vanderzande, Carlo, 1998. "Sandpiles on a Sierpinski gasket," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 256(3), pages 533-546.
    3. 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.
    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.
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