On some growth models with a small parameter

Abstract

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 ℤd in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate ɛ≦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 (as time goes to infinity) and these speeds determine an asymptotic shape in the usual sense. It is proven that as ɛ tends to 0, the asymptotic speeds scale as ɛ1/d and the asymptotic shape, when renormalized by dividing it by ɛ1/d , converges to a cube. Other similar models which are partially oriented are also studied.