Correlations, mean-field properties, and scaling of a one-dimensional sandpile model

We present a general relationship between different scaling exponents for the one-dimensional sandpile problem to describe the self-adjusting of the slope of the sandpile. We solve the mean-field theory for this model, assuming that there is no correlation between the sizes of neighbor clusters. The mean-field theory does not give the correct exponents, since the clusters are strongly correlated. We characterize these correlations, identify the functional form of the cluster distribution function, and show how the multifractal scaling for averaged quantities arises from this form.