Singularities and Avalanches in Interface Growth with Quenched Disorder

A simple model for an interface moving in a disordered medium is presented. The model exhibits a transition between the two universality classes of interface growth phenomena. Using this model, it is shown that the application of constraints to the local slopes of the interface produces avalanches of growth, that become relevant in the vicinity of the depinning transition. The study of these avalanches reveals a singular behavior that explains a recently observed singularity in the equation of motion of the interface.