Integer superharmonic matrices on the -lattice
arXiv:2110.07556 · doi:10.1016/j.aim.2023.109400
Abstract
We prove that the set of quadratic growths achievable by integer superharmonic functions on the -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 -lattice.
arXiv comment85 pages, 34 figures; 2 tables; v2 minor corrections