Pattern formation in growing sandpiles

Adding grains at a single site on a flat substrate in the abelian sandpile models produces beautiful complex patterns. These patterns show proportionate growth where all spatial features increase at equal rate, keeping the overall shape unchanged. I will discuss in detail, the pattern produced in the model defined 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. This is a good example of analytically tractable complex patterns generated using simple local rules. I will present an exact characterization of the asymptotic pattern, which involves some intriguing connection to mathematics of discrete analytic functions. I will also discuss how this pattern and its growth rate changes in presence of absorbing sites.