The Upper Critical Dimension of the Abelian Sandpile Model

Abstract

The existing estimation of the upper critical dimension of the Abelian Sandpile Model is based on a qualitative consideration of avalanches as self-avoiding branching processes. We find an exact representation of an avalanche as a sequence of spanning subtrees of two-component spanning trees. Using equivalence between chemical paths on the spanning tree and loop-erased random walks, we reduce the problem to determination of the fractal dimension of spanning subtrees. Then the upper critical dimension du=4 follows from Lawler's theorems for intersection probabilities of random walks and loop-erased random walks.