Anchored advected interfaces, Oslo model, and roughness at depinning

There is a plethora of 1-dimensional advected systems with an absorbing boundary: the Toom model of anchored interfaces, the directed exclusion process where in addition to diffusion particles and holes can jump over their right neighbor, simple diffusion with advection, and Oslo sandpiles. All these models share a roughness exponent of $\zeta =1\mathrm{/}4$, while the dynamic exponent $z$ varies, depending on the observable. We show that for the first three models $z=1$, $z=2$, and $z=1\mathrm{/}2$ are realized, depending on the observable. The Oslo model is apart with a conjectured dynamic exponent of $z=10\mathrm{/}7$. Since the height in the latter is the gradient of the position of a disordered elastic string, this shows that $\zeta =5\mathrm{/}4$ for a driven elastic string at depinning.