The Apollonian structure of integer superharmonic matrices

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 that determines the continuum scaling limit of the Abelian sandpile on the lattice $\mathbb{Z}^2$.

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Authors

Lionel Levine

Cornell University, Ithaca, NY

Wesley Pegden

Carnegie Mellon University, Pittsburgh, PA

Charles K. Smart

The University of Chicago, Chicago, IL