  Interfaces advancing through random media represent a number of different
problems in physics, biology and other disciplines. Here, we study the
pinning/depinning transition of the prototypical non-equilibrium interfacial
model, i.e. the Kardar-Parisi-Zhang equation, advancing in a disordered medium.
We analyze separately the cases of positive and negative non-linearity
coefficients, which are believed to exhibit qualitatively different behavior:
the positive case shows a continuous transition that can be related to
directed-percolation-depinning while in the negative case there is a
discontinuous transition and faceted interfaces appear. Some studies have
argued from different perspectives that both cases share the same universal
behavior. Here, by using a number of computational and scaling techniques we
shed light on this puzzling situation and conclude that the two cases are
intrinsically different.
