H. Kesten, R. H. Schonmann
On some growth models with a small parameter
(98K, AMSTeX)

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.