  The dynamics of sandpile models are mapped to discrete interface equations.
We study in detail the Bak-Tang-Wiesenfeld model, a stochastic model with
random thresholds, and the Manna model. These are, respectively,
discretizations of the quenched Edwards-Wilkinson equation with columnar,
point-like and correlated noise, with the constraint that the interface
velocity is either zero or exactly one. The constraint, embedded in the
sandpile rules, gives rise to another noise component. This term has for the
Bak-Tang-Wiesenfeld model long-range on-site correlations and reveals that with
open boundary conditions there is no spatial translational invariance.
