Kinetic roughening of Laplacian fronts

The kinetic fluctuations of a stable interface driven by the gradient of a Laplacian field are investigated. In three and more dimensions the interface width is finite. In two dimensions the width diverges logarithmically with time and system size. Its scaling form is derived in agreement with simulations of diffusion-limited erosion (anti diffusion-limited aggregation). A crossover to algebraic roughness with an extended intermediate scaling regime is predicted for diffusion with a drift towards the interface. Capillary effects are discussed in relation to recent experiments on fluid displacement in porous media.