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Cusps, self-organization, and absorbing states

Elastic interfaces embedded in (quenched) random media exhibit meta-stability and stick-slip dynamics. These non-trivial dynamical features have been shown to be associated with cusp singularities of the coarse-grained disorder correlator. Here we show that annealed systems with many absorbing states and a conservation law but no quenched disorder exhibit identical cusps. On the other hand, similar non-conserved systems in the directed percolation class, are also shown to exhibit cusps, but of a different type. These results are obtained both by a recent method to explicitly measure disorder correlators and by defining an alternative new protocol, inspired by self-organized criticality, which opens the door to easily accessible experimental realizations.