Avalanches and Waves in the Abelian Sandpile Model

TL;DR

This study numerically investigates wave dynamics in the 2D Abelian sandpile model, revealing that previous assumptions about wave size relations are invalid, with significant implications for understanding critical behavior.

Contribution

The paper challenges prior theoretical predictions by showing that the assumed scaling relation for wave size differences does not hold in the 2D Abelian sandpile model.

Findings

Average size difference <Δs> is negative and diverges with system size.

Waves violating the assumed bound significantly influence the dynamics.

Contradicts previous proposed scaling relations for wave sizes.

Abstract

We numerically study avalanches in the two dimensional Abelian sandpile model in terms of a sequence of waves of toppling events. Priezzhev et al [PRL 76, 2093 (1996)] have recently proposed exact results for the critical exponents in this model based on the existence of a proposed scaling relation for the difference in sizes of subsequent waves, $\Delta s =s_{k}- s_{k+1}$, where the size of the previous wave $s_{k}$ was considered to be almost always an upper bound for the size of the next wave $s_{k+1}$. Here we show that the significant contribution to $\Delta s$ comes from waves that violate the bound; the average $<\Delta s(s_{k})>$ is actually negative and diverges with the system size, contradicting the proposed solution.