Integer superharmonic matrices on the $F$lattice
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.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07556
 Bibcode:
 2021arXiv211007556B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Number Theory;
 Mathematics  Probability
 EPrint:
 83 pages, 34 figures