Directed percolation depinning models: Evolution equations

We present the microscopic equation for the growing interface with quenched noise for the model first presented by Buldyrev et al. [Phys. Rev. A 45, R 8313 (1992)]. The evolution equation for the height, the mean height, and t he roughness are reached in a simple way. The microscopic equation allows u s to express these equations in two contributions: the contact and the loca l one. We compare these two contributions with the ones obtained for the Ta ng and Leschhorn model [Phys. Rev A 45, R8309 (1992)] by Braunstein et al. [Physica A, 266, 308 (1999)]. Even when the microscopic mechanisms are quie t different in both models, the two contributions are qualitatively similar . An interesting result is that the diffusion contribution, in the Tang and Leschhorn model, and the contact one, in the Buldyrev model, leads to an i ncrease of the roughness near the criticality.