Pattern formation in growing sandpiles

摘要

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 root N. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern.