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arXiv · 1309.3267

The Apollonian structure of integer superharmonic matrices

Lionel Levine·Wesley Pegden·Charles K. Smart
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Abstract

We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely characterizes the PDE which determines the continuum scaling limit of the Abelian sandpile on the lattice $\mathbb{Z}^2$.

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Lionel Levine, Wesley Pegden, Charles K. Smart. 2013-09-12. The Apollonian structure of integer superharmonic matrices. https://arxiv.org/abs/1309.3267

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