Abstract
We study the two-dimensional Abelian Sandpile Model on a square lattice of linear size L. We introduce the notion of avalanche’s fine structure and compare the behavior of avalanches and waves of toppling. We show that according to the degree of complexity in the fine structure of avalanches, which is a direct consequence of the intricate superposition of the boundaries of successive waves, avalanches fall into two different categories. We propose scaling ansätz for these avalanche types and verify them numerically. We find that while the first type of avalanches (α) has a simple scaling behavior, the second complex type (β) is characterized by an avalanche-size dependent scaling exponent. In particular, we define an exponent γ to characterize the conditional probability distribution functions for these types of avalanches and show that γ α = 0.42, while 0.7 ≤ γ β ≤ 1.0 depending on the avalanche size. This distinction provides a framework within which one can understand the lack of a consistent scaling behavior in this model, and directly addresses the long-standing puzzle of finite-size scaling in the Abelian sandpile model.
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The reason for this is the saturation of average number of waves as a function of area in the case of type α avalanches, so that the exponent γ introduce in reference [30] is nearly zero for intermediate and large α avalanches, see reference [53]
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Abdolvand, A., Montakhab, A. Scaling and complex avalanche dynamics in the Abelian sandpile model. Eur. Phys. J. B 76, 21–30 (2010). https://doi.org/10.1140/epjb/e2010-00164-8
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DOI: https://doi.org/10.1140/epjb/e2010-00164-8