Apollonian structure in the Abelian sandpile
Analysis of PDEs
2014-05-23 v2 Statistical Mechanics
Combinatorics
Number Theory
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.
Cite
@article{arxiv.1208.4839,
title = {Apollonian structure in the Abelian sandpile},
author = {Lionel Levine and Wesley Pegden and Charles K. Smart},
journal= {arXiv preprint arXiv:1208.4839},
year = {2014}
}
Comments
27 Pages, 7 Figures