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 d u=4 follows from Lawler's theorems for intersection probabilities of random walks and loop-erased random walks.
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Priezzhev, V.B. The Upper Critical Dimension of the Abelian Sandpile Model. Journal of Statistical Physics 98, 667–684 (2000). https://doi.org/10.1023/A:1018619323983
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DOI: https://doi.org/10.1023/A:1018619323983