The Apollonian structure of integer superharmonic matrices

Published 12 Sep 2013 in math.AP, math.NT, and math.PR | (1309.3267v4)

Abstract: 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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