Expansion and Contraction of Avalanches in 2D Abelian Sandpile

TL;DR

This paper analyzes large-scale simulations of avalanches in the 2D Abelian sandpile model, comparing different cluster decompositions, confirming the existence of a contraction exponent, and providing detailed exponent values for wave statistics.

Contribution

It introduces auxiliary critical exponents and confirms the contraction exponent, offering refined arguments for wave exponent values in 2D Abelian sandpile avalanches.

Findings

Confirmed the existence of the contraction exponent.

Compared statistical properties of avalanche decompositions.

Provided exact values for wave-related exponents.

Abstract

We present a detailed analysis of large scale simulations of avalanches in the 2D Abelian sandpile model. We compare statistical properties of two different decompositions of avalanches into clusters of topplings and waves of topplings. Auxiliary critical exponents are introduced and the existence of the exponent governing the contraction of avalanches claimed in our previous work [Priezzhev et al, PRL 76,2093 (1996)] is confirmed. We also give more elaborated argumentation for the exact values of the exponents characterizing the statistics of waves.