A solvable model of interface depinning in random media

We study the mean-field version of a model proposed by Leschhorn to describe the depinning transition of interfaces in random media. We show that evolution equations for the distribution of forces felt by the interface sites can be written down directly for an infinite system. For a flat distribution of random local forces the value of the depinning threshold can be obtained exactly. In the case of parallel dynamics (all unstable sites move simultaneously), due to the discrete character of the allowed interface heights, the motion of the center of mass is non-uniform in time in the moving phase close to the threshold and the mean interface velocity vanishes with a square-root singularity.