Avalanches and waves in the Abelian sandpile model

We numerically study avalanches in the two-dimensional Abelian sandpile model in terms of a sequence of waves of toppling events. Priezzhev et al. [Phys. Rev. Lett. 76, 2093 (1996)] have recently proposed exact results for the critical exponents in this model based on the existence of a proposed scaling relation for the difference in sizes of subsequent waves, $\Delta {s=s}_k-s_{k+1\mathrm{}},$ where the size of the previous wave $s_k$ was considered to be almost always an upper bound for the size of the next wave $s_{k+1\mathrm{}}.$ Here we show that the significant contribution to $\Delta s$ comes from waves that violate the bound; the average $〈\Delta {s(s}_k)〉$ is actually negative and diverges with the system size, contradicting the proposed solution.