94-24 H. Kesten, R. H. Schonmann

On some growth models with a small parameter (98K, AMSTeX) Jan 26, 94

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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 $\Z^{d}$ in which each site which is not yet occupied becomes occupied at rate 1 if it has at least two occupied neighbors, at rate $\varepsilon \le 1$ 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 $\varepsilon$ tends to $0$, the asymptotic speeds scale as $\varepsilon^{1/d}$ and the asymptotic shape, when renormalized by dividing it by $\varepsilon^{1/d}$, converges to a cube. Other similar models which are partially oriented are also studied.

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