Rare events and breakdown of simple scaling in the Abelian sandpile

TL;DR

This paper investigates how rare, large avalanches in the 2D Abelian sandpile cause deviations from simple scaling laws, revealing multifractal behavior and the dominance of rare events in the system's statistics.

Contribution

It demonstrates that rare events lead to multifractal scaling in the Abelian sandpile and clarifies their role in the breakdown of simple finite size scaling.

Findings

Large avalanches dominate the statistics in the thermodynamic limit.

Multifractal scaling replaces simple finite size scaling.

Effective exponents can be derived from numerical and exact results.

Abstract

Due to intermittency and conservation, the Abelian sandpile in 2D obeys multifractal, rather than finite size scaling. In the thermodynamic limit, a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution. Thus, rare events shape most of the scaling pattern and preserve a meaning for effective exponents, which can be determined on the basis of numerical and exact results.