Breakdown of Simple Scaling in Abelian Sandpile Models in One Dimension

TL;DR

This paper investigates the avalanche size distribution in decorated one-dimensional Abelian sandpile models, revealing that their scaling behavior deviates from simple forms and involves a combination of two distinct scaling functions.

Contribution

It demonstrates that the avalanche size distribution in these models cannot be described by a single scaling law but by a linear combination of two scaling functions, highlighting a breakdown of simple scaling.

Findings

Distribution involves two scaling functions, not one.

Scaling form is a linear combination of $f_1$ and $f_2$.

Qualitative difference from simple linear chain models.

Abstract

We study the abelian sandpile model on decorated one dimensional chains. We determine the structure and the asymptotic form of distribution of avalanche-sizes in these models, and show that these differ qualitatively from the behavior on a simple linear chain. We find that the probability distribution of the total number of topplings $s$ on a finite system of size $L$ is not described by a simple finite size scaling form, but by a linear combination of two simple scaling forms $Prob_L(s) = 1/L f_1(s/L) + 1/L^2 f_2(s/L^2)$, for large $L$, where $f_1$ and $f_2$ are some scaling functions of one argument.