Apollonian structure in the Abelian sandpile |
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Authors: | Lionel Levine Wesley Pegden Charles K. Smart |
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Affiliation: | 1.Department of Mathematics,Cornell University,Ithaca,USA;2.Carnegie Mellon University,Pittsburgh,USA;3.Massachusetts Institute of Technology,Cambridge,USA |
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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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