Abelian sandpile model on the Bethe lattice

Abstract

The authors study Bak, Tang and Wiesenfeld's Abelian sandpile model (1987) of self-organised criticality on the Bethe lattice. Exact expressions for various distribution functions including the height distribution at a site and the joint distribution of heights at two sites separated by an arbitrary distance are obtained. They also determine the probability distribution of the number of distinct sites that topple at least once, the number of toplings at the origin and the total number of toplings in an avalanche. The probability that an avalanche consists of more than n toplings varies as n-1/2 for large n. The probability that its duration exceeds T decreases as 1/T for large T. These exponents are the same as for the critical percolation clusters in mean field theory.