Structure of avalanches and breakdown of simple scaling in the Abelian sandpile model in one dimension

We study the Abelian sandpile model on decorated one dimensional chains. We show that there are two types of avalanches, and determine the effects of finite, though large, system size L on the asymptotic form of distributions of avalanche sizes, and show that these differ qualitatively from the behavior on a simple linear chain. For large L, we find that the probability distribution of the total number of topplings s 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/${\mathit{L}}^2$), where ${\mathit{f}}_1$ and ${\mathit{f}}_2$ are nonuniversal scaling functions of one argument.