Universality classes for deterministic surface growth

Abstract

We study the growth of a surface through deterministic local rules. A scaling theory for the generalized deterministic Kardar-Parisi-Zhang equation ${\ensuremath{\partial}}_{t}$h=D \ensuremath{\Delta}h+\ensuremath{\lambda}\ensuremath{\Vert}\ensuremath{\nabla}h${\ensuremath{\Vert}}^{\ensuremath{\beta}}$, with \ensuremath{\beta}\ensuremath{\ge}1, is developed. A one-dimensional surface model, which corresponds to \ensuremath{\beta}=1, is solved exactly. It can be obtained as a limiting case of ballistic deposition, or as the deterministic limit of the Eden model. We determine the scaling exponents, the correlation functions, and the skewness of the surface. We point out analogies to the Burgers equation (\ensuremath{\beta}=2), for which such detailed properties are not known.