Scaling of many-particle correlations in a dissipative sandpile

Abstract

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 can thereby be solved exactly, and an explicit expression is given for the asymptotic form of the solution for an infinite as well as for a semi-infinite lattice in the horizontal direction. Non-trivial scaling is found in both cases. This conditional probability of the sandpile model is shown to be equal to a K-spin correlation function of the Heisenberg XX spin chain, and the sandpile problem is also shown to be equivalent to the `random-turns' version of vicious walkers.