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 which consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that the dissipative sandpiles' correlation length exponent \nu 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 results that \nu=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.