Dissipative Abelian sandpiles and random walks

Abstract

We show that the dissipative Abelian sandpile on a graph L can be related to a random walk on a graph that consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that the correlation length exponent nu of the dissipative sandpiles always equals 1/d(w) where d(w) is the fractal dimension of the random walker. This leads to a new understanding of the known result that v = 1/2 on any Euclidean lattice. Our result is, however, more general, and as an example we also present exact data for finite Sierpinski gaskets, which fully confirm our predictions.