Pinning by rare defects and effective mobility for elastic interfaces in high dimensions

The existence of a depinning transition for a high dimensional interface in a weakly disordered medium is controversial. Following Larkin arguments and a perturbative expansion, one expects a linear response with a renormalized mobility ${\mu}_{\text{eff}}$ . In this paper, we compare these predictions with the exact solution of a fully connected model, which displays a finite critical force $f_c$. At small disorder, we unveil an intermediary linear regime for $f_c < f < 1$ characterized by the renormalized mobility ${\mu}_{\text{eff}}$. Our results suggest that in high dimension the critical force is always finite and determined by the effect of rare impurities that is missed by the perturbative expansion. However, the perturbative expansion correctly describes an intermediate regime that should be visible at small disorder.