Emergence and Complexity in Theoretical Models of Self-Organized Criticality

Adding grains at a single site on a flat substrate in the abelian sandpile models produces beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The model shows proportionate growth and the size of the pattern formed by adding N grains scales as √ N. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features in the pattern.