The Upper Critical Dimension of the Abelian Sandpile Model

1999/04/04 by V. B. Priezzhev, Priezzhev, V. B.
Earth and Planetary Sciences · Engineering · Physics and Astronomy · #FOS: Physical sciences #Geological formations and processes #Hydraulic Fracturing and Reservoir Analysis #Hydrocarbon exploration and reservoir analysis #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech

paper · pdf · doi:10.48550/arxiv.cond-mat/9904054

16 pages

arxiv created 1999/04/04 · openalex publication_date 1999/04/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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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