Integer superharmonic matrices on the $F$-lattice
Analysis of PDEs
2023-11-07 v2 Number Theory
Probability
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.
Cite
@article{arxiv.2110.07556,
title = {Integer superharmonic matrices on the $F$-lattice},
author = {Ahmed Bou-Rabee},
journal= {arXiv preprint arXiv:2110.07556},
year = {2023}
}
Comments
85 pages, 34 figures; 2 tables; v2 minor corrections