Apollonian structure in the Abelian sandpile
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39_2016_Article_358.pdf
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1.22 MB
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Author(s)
Levine, Lionel
Pegden, Wesley
Smart, Charles
Date Issued
February 2016
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
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00039-016-0358-7
