The Apollonian structure of integer superharmonic matrices

TL;DR

This paper reveals that the set of quadratic growths of integer superharmonic functions on a 2D lattice forms an Apollonian circle packing, characterizing the PDE governing the sandpile's continuum limit.

Contribution

It establishes a novel geometric structure (Apollonian packing) for superharmonic functions and links it to the PDE describing the sandpile's scaling limit.

Findings

Quadratic growths form an Apollonian circle packing

Characterizes the PDE for the Abelian sandpile continuum limit

Provides a geometric framework for superharmonic functions

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 $\mathbb{Z}^2$.