Apollonian structure in the Abelian sandpile
Author(s)Levine, Lionel; Pegden, Wesley; Smart, Charles
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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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Geometric and Functional Analysis
Springer International Publishing
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.
Author's final manuscript