Dynamic scaling of growing interfaces

A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burger's equation and to a random directed polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions