The Sandpile Group on a Hexagonal Grid

Abstract

We introduce the notion of an abelian sandpile, including the basic definitions, its use as a model, and some of the foundational theorems and observations. We describe a constructive method for reducing redundant computations when the underlying graph has one or more symmetries. We motivate the analysis of a particular two-dimensional model: the hexagonal tiling of R^2, and extend a convergence result for the integer lattice Z^d to this hexagonal grid, using discrete approximation of partial differential equation methods. We end with some asymptotic conjectures on this grid.