Pattern formation in growing sandpiles

Understand

Adding grains at a single site on a flat substrate in the Abelian sandpile models produce 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 size of the pattern formed scales with the number of grains added $N$ as $\sqrt{N}$.
• We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern.