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Apollonian structure in the Abelian sandpile

Author(s)
Levine, Lionel; Pegden, Wesley; Smart, Charles
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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.
Date issued
2016-02
URI
http://hdl.handle.net/1721.1/106972
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Geometric and Functional Analysis
Publisher
Springer International Publishing
Citation
Levine, Lionel, Wesley Pegden, and Charles K. Smart. “Apollonian Structure in the Abelian Sandpile.” Geometric and Functional Analysis 26, no. 1 (February 2016): 306–336.
Version: Author's final manuscript
ISSN
1016-443X
1420-8970

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