The Upper Critical Dimension of the Abelian Sandpile Model

TL;DR

This paper determines the upper critical dimension of the Abelian Sandpile Model as exactly 4 by connecting avalanches to spanning trees and loop-erased random walks, providing a rigorous mathematical proof.

Contribution

It offers an exact mathematical derivation of the upper critical dimension for the Abelian Sandpile Model, improving upon previous qualitative estimates.

Findings

Upper critical dimension is exactly 4.

Avalanches are represented as spanning sub-trees.

Uses Lawler's theorems to establish the result.

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.