The Upper Critical Dimension of the Abelian Sandpile Model

Details The Upper Critical Dimension of the Abelian Sandpile Model The Upper Critical Dimension of the Abelian Sandpile Model

1999-04-04

arxiv.org/abs/cond-mat/9904054v1

Physics

Condensed Matter

Statistical Mechanics

16 pages

Scientific paper

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 sub-trees 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 sub-trees. 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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