Apollonian structure in the Abelian sandpile

2012/08/23 by Levine, Lionel, Pegden, Wesley, Smart, Charles K. · 2 citations
#35R35 #60K35 #Analysis of PDEs (math.AP) #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Number Theory (math.NT) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.1208.4839

Abstract

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.

Cited by

• The odometer in subcritical activated random walk
• The limit shape of the Leaky Abelian Sandpile Model on isoradial graphs

Related

• The Apollonian structure of integer superharmonic matrices
• The Abelian Sandpile Model on Fractal Graphs
• Evolutive sandpiles
• The Upper Critical Dimension of the Abelian Sandpile Model
• Sandpiles on the Vicsek fractal explode with probability 1/4