Patterns formed by addition of grains to only one site of an abelian sandpile

We study the patterns formed by adding N sand grains at a single site in the abelian sandpile model on an infinite square lattice, when in the initial configuration all sites have same height. When this height is 2, stable heights being 0–3, we show that the perturbed region is a square, whose length increases as N for large N. If all lengths are rescaled by N, the pattern tends to a limiting pattern, in which the locally averaged height takes piece wise constant rational values. We study the structure of these patterns, and also other initial configurations. We introduce a toppling function that fully describes these we show that it is piecewise quadratic and determine its form, as well as the relation with the height configurations of the pattern.