Scaling properties of trough densities in sandpile models

We characterize the self-organized criticality of a class of one-dimensional sandpile models (limited local models) in terms of the scaling properties of troughs whose spatial distribution controls the size of avalanches. We establish, both by a mean-field approximation and simulations, that the trough density ρ_k is exponentially small in the depth k and that the algebraic decay of the local, overall trough density is governed by the diffusion singularity in the continuum limit studied by Carlson et al. [Phys. Rev. Lett. 65, 2547 (1990)]. Multiscaling is seen to arise from the correlations neglected in the approximation.