Fine Structure of Avalanches in the Abelian Sandpile Model

TL;DR

This paper investigates the detailed internal structure of avalanches in the Abelian Sandpile Model, revealing two distinct categories with different scaling behaviors, and proposes a new framework to understand their complex scaling properties.

Contribution

It introduces the concept of avalanche fine structure, classifies avalanches into two types based on complexity, and proposes scaling laws verified through numerical analysis.

Findings

Two avalanche types with distinct scaling behaviors

Complex avalanches exhibit size-dependent scaling exponents

Framework explains variability in scaling in the model

Abstract

We study the two-dimensional Abelian Sandpile Model on a square lattice of linear size L. We introduce the notion of avalanche's fine structure and compare the behavior of avalanches and waves of toppling. We show that according to the degree of complexity in the fine structure of avalanches, which is a direct consequence of the intricate superposition of the boundaries of successive waves, avalanches fall into two different categories. We propose scaling ans\"{a}tz for these avalanche types and verify them numerically. We find that while the first type of avalanches has a simple scaling behavior, the second (complex) type is characterized by an avalanche-size dependent scaling exponent. This provides a framework within which one can understand the failure of a consistent scaling behavior in this model.