The Apollonian structure of integer superharmonic matrices

2013/09/12 by Levine, Lionel, Pegden, Wesley, Smart, Charles K. · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1309.3267

Abstract

We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice ℤ² 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 ℤ².

Cited by

• Scaling and partial universality of the height zero probability in the 2D Abelian sandpile
• The odometer in subcritical activated random walk

Related

• Integer superharmonic matrices on the F-lattice
• Apollonian structure in the Abelian sandpile
• Apollonian Packings and Kac-Moody Root Systems
• The sensual Apollonian circle packing
• Apollonian Circle Packings: Number Theory