Breakdown of simple scaling in Abelian sandpile models in one dimension

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 ${\mathrm{Prob}}_{\mathit{L}}$(s)=(1/L)${\mathit{f}}_1$(s/L)+(1/${\mathit{L}}^2$)${\mathit{f}}_2$(s/${\mathit{L}}^2$), for large L, where ${\mathit{f}}_1$ and ${\mathit{f}}_2$ are some scaling functions of one argument.