ON SOME GROWTH-MODELS WITH A SMALL-PARAMETER

We consider the behavior of the asymptotic speed of growth and the asy mptotic 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 oc cupied at rate 1 if it has at least two occupied neighbors, at rate ep silon less than or equal to 1 if it has exactly 1 occupied neighbor an d, of course, at rate 0 if it has no occupied neighbor. Occupied sites remain occupied forever. Starting from a single occupied site, this m odel has asymptotic speeds of growth in each direction (as time goes t o infinity) and these speeds determine an asymptotic shape in the usua l. sense. It is proven that as epsilon tends to 0, the asymptotic spee ds scale as epsilon 1/d and the asymptotic shape, when renormalized by dividing it by epsilon(1/d), converges to a cube. Other similar model s which are partially oriented are also studied.