# 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 $\mathbb{Z}^2$

## Authors

Lionel Levine

Cornell University, Ithaca, NY 14853

Wesley Pegden

Carnegie Mellon University, Pittsburgh, PA 15213

Charles K. Smart

University of Chicago, Chicago, IL 60637