Abstract
We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ℤd. Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h≤2d−2, the diameter of the set of sites that topple has order n 1/d. This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification.
We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d−1. On the other hand, we show that if the background height 2d−2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).
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References
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)
Björner, A., Lovász, L., Shor, P.: Chip-firing games on graphs. Eur. J. Comb. 12(4), 283–291 (1991)
Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613 (1990)
Diaconis, P., Fulton, W.: A growth model, a game, an algebra, Lagrange inversion, and characteristic classes. Rend. Semin. Mat. Univ. Pol. Torino 49(1), 95–119 (1991)
van Enter, A.C.D.: Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys. 48, 943–945 (1987)
Fey, A., Redig, F.: Limiting shapes for deterministic centrally seeded growth models. J. Stat. Phys. 130(3), 579–597 (2008)
Fey-den Boer, A., Redig, F.: Organized versus self-organized criticality in the abelian sandpile model. Markov Processes Relat. Fields 11(3), 425–442 (2005)
Fey-den Boer, A., Meester, R., Redig, F.: Stabilizability and percolation in the infinite volume sandpile model. Ann. Probab. 37(2), 654–675 (2009)
Haase, C., Musiker, G., Yu, J.: Linear systems on tropical curves (2009). http://arxiv.org/abs/0909.3685
Levine, L., Peres, Y.: Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile. Potential Anal. 30, 1–27 (2009)
Meester, R., Redig, F., Znamenski, D.: The abelian sandpile model, a mathematical introduction. Markov Processes Relat. Fields 7, 509–523 (2001)
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Fey, A., Levine, L. & Peres, Y. Growth Rates and Explosions in Sandpiles. J Stat Phys 138, 143–159 (2010). https://doi.org/10.1007/s10955-009-9899-6
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DOI: https://doi.org/10.1007/s10955-009-9899-6