Interface motion in random media

We study the pinning phase transition for discrete surface dynamics in random environments. A renormalization procedure is devised to prove that the interface moves with positive velocity under a finite size condition. This condition is then checked for different examples of microscopic dynamics to illustrate the flexibility of the method. We show in our examples the existence of a phase transition for various models, including high dimensional interfaces, dependent environments and environments with arbitrarily deep obstacles. Finally, our ballisticity criterion is proved to be valid up to the critical threshold for a Lipschitz interface model.