Multiple-grain dissipative sandpiles

The two-dimensional directed sandpile with dissipation is transformed into a $(1+1)$-dimensional problem with discrete space and continuous “time”. The master equation for the conditional probability that $K$ grains preserve their initial order during an avalanche is solved exactly. Explicit expressions for asymptotic forms of solutions are given for the cases of the infinite and semi-infinite lattices. A nontrivial scaling is found in both cases.