Integer superharmonic matrices on the $F$-lattice

TL;DR

This paper characterizes the set of quadratic growths of integer superharmonic functions on the $F$-lattice, revealing a circle packing structure and confirming a conjecture about the Abelian sandpile's scaling limit.

Contribution

It proves a conjecture by Smart (2013) and fully describes the scaling limit of the Abelian sandpile on the $F$-lattice, introducing a novel geometric structure.

Findings

Quadratic growths form an overlapping circle packing

Constructs recurrent functions for each rational point on a hyperbola

Completes the description of the Abelian sandpile's scaling limit

Abstract

We prove that the set of quadratic growths achievable by integer superharmonic functions on the $F$-lattice, a periodic subgraph of the square lattice with oriented edges, has the structure of an overlapping circle packing. The proof recursively constructs a distinct pair of recurrent functions for each rational point on a hyperbola. This proves a conjecture of Smart (2013) and completely describes the scaling limit of the Abelian sandpile on the $F$-lattice.