Lionel Levine, Wesley Pegden, Charles K. Smart
We state a conjecture relating integer-valued superharmonic functions on $\mathbb{Z}^2$ to an Apollonian circle packing of $\mathbb{R}^2$. The conjecture is motivated by the Abelian sandpile process, which 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. Our conjecture implies that the Sandpile PDE recently shown to characterize the continuum limit of the sandpile is equivalent to the Apollonian PDE, and we use the special geometric structure of the latter to prove that it admits certain fractal solutions. Boundary condition evidence from finite sandpiles suggest that these solutions exactly correspond to regions of the limiting sandpile, leading to precise geometric conjectures on the Abelian sandpile's fractal behavior.
View original:
http://arxiv.org/abs/1208.4839
No comments:
Post a Comment