Exact scaling in surface growth with power-law noise

We investigate a continuum formulation of surface growth following the Kardar-Parisi-Zhang equation [Phys. Rev. Lett. 56, 889 (1986)] with a power-law distribution of the magnitudes of regional advances. This formulation describes Zhangs ballistic-deposition model [J. Phys. (Paris) 51, 2129 (1990)] with power-law noise and possibly recent fluid-displacement experiments. Our exact theory predicts a transition of the scaling behavior from power-law-noise domination to a Gaussian-noise regime as the power increases. An apparent contradiction with previous simulations is due to a logarithmic correction to the scaling at the transition and to anomalous-growth effects. Analogous scaling behaviors are derived for the Edwards-Wilkinson model [Proc. R. Soc. London Ser. A 381, 17 (1982)] with power-law noise. Our results are supported by simulations.