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

56 pages, many figures. Typo in T1.1 corrected (2 -> 0)