Growth of a driven interface in isotropic and anisotropic random media

We introduce a simple stochastic model for a driven interface in a random medium, in which we can control the degree of the anisotropy of a random medium. When there is no anisotropy of a random medium, the motion of a growing interface in our model can be well described by the quenched Edwards-Wilkinson equation. When there is anisotropy of a random medium, however, the motion of a growing interface can be described by the quenched Kardar-Parisi-Zhang (KPZ) equation. In the two interfaces, apart from one growing in an isotropic medium and the other growing in an anisotropic medium, the growth rule of our model is the same. Our results support the fact that the anisotropy of a random medium is a source of the KPZ nonlinearity.