Geodesic
Multiple-grain dissipative sandpiles
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 5-18

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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. 
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     author = {N. M. Bogoliubov and A. G. Pronko and J. Timonen},
     title = {Multiple-grain dissipative sandpiles},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a0/}
}
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N. M. Bogoliubov; A. G. Pronko; J. Timonen. Multiple-grain dissipative sandpiles. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 5-18. http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a0/