The Apollonian structure of integer superharmonic matrices
classification
🧮 math.AP
math.NTmath.PR
keywords
apollonianlatticemathbbstructuresuperharmonicabelianattainablecharacterizes
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We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely characterizes the PDE which determines the continuum scaling limit of the Abelian sandpile on the lattice $\mathbb{Z}^2$.
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