From ballistic deposition to the Kardar-Parisi-Zhang equation through a limiting procedure

We show a direct connection between the ballistic deposition and the Kardar-Parisi-Zhang (KPZ) equation. We derive the KPZ equation from the ballistic deposition models, using an important limiting procedure. The cellular automaton rule is transformed into an integrable difference-difference equation through the limiting procedure. By applying the perturbation method to the difference-difference equation, the difference-difference equation is reduced to the KPZ equation through the Burgers equation. We apply the procedure to several types of ballistic deposition models.