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arXiv · 1208.4839

Apollonian structure in the Abelian sandpile

Lionel Levine·Wesley Pegden·Charles K. Smart
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Abstract

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.

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Lionel Levine, Wesley Pegden, Charles K. Smart. 2012-08-23. Apollonian structure in the Abelian sandpile. https://arxiv.org/abs/1208.4839

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