Inverse avalanches in the Abelian sandpile model

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 ${\mathit{s}}_1$, the number of ‘‘untopplings’’ in the first inverse avalanche. For a square lattice, we determine Prob(${\mathit{s}}_1$) exactly for ${\mathit{s}}_1$=0, 1, 2, and 3. For large ${\mathit{s}}_1$, we show that Prob(${\mathit{s}}_1$) varies as ${\mathit{s}}_1^{\mathrm{-}11/8}$. 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.