Breakdown of simple scaling in abelian sandpile models in one dimension

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)f₁(s/L)+(1/L²)f₂(s/L²), for large L, where f₁ and f₂ are some scaling functions of one argument.