We report the generalized spatial discretization of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions. We solve exactly the steady state probability density function for the discrete heights of the interface, for any discretization scheme. We show that the discretization prescription is a consequence of each particular model. We derive the discretization prescription of the KPZ equation for the ballistic deposition model.

• Received 21 October 2004

DOI:https://doi.org/10.1103/PhysRevE.72.017701