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arXiv · 2110.07556

Integer superharmonic matrices on the $F$-lattice

Ahmed Bou-Rabee
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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.

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Ahmed Bou-Rabee. 2021-10-14. Integer superharmonic matrices on the $F$-lattice. https://arxiv.org/abs/2110.07556

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