On Some Growth Models with a Small Parameter

We consider the behavior of the asymptotic speed of growth and the asymptotic shape in some growth models, when a certain parameter becomes small. The basic example treated is the variant of Richardson's growth model on Z d in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate " 1 if it has exactly 1 occupied neighbor and, of course, at rate 0 if it has no occupied neighbor. Occupied sites remain occupied forever. Starting from a single occupied site, this model has asymptotic speeds of growth in each direction and these speeds determine an asymptotic shape in the usual sense. It is proven that as " tends to 0, the asymptotic speeds scale as " 1=d and the asymptotic shape, when renormalized by dividing it by " 1=d , converges to a cube. Other similar models which are partially oriented are also studied.