Inverse avalanches in the abelian sandpile model

Abstract

We define and study the inverse of particle addition process in the Abelian sandpile model. We show how to obtain the unique recurrent configuration corresponding to a single particle deletion by a sequence of operations called inverse avalanches. We study the probability distribution of s₁, the number of "untopplings" in the first inverse avalanche. For a square lattice, we determine Prob(s₁) exactly for s₁=0, 1, 2, and 3. For large s₁, we show that Prob(s₁) varies as s₁^-11/8. In the direct avalanches, this is related to the probability distribution of the number of sites which topple as often as the site where the particle was added. These results are verified by numerical simulations.