 9424 H. Kesten, R. H. Schonmann
 On some growth models with a small parameter
(98K, AMSTeX)
Jan 26, 94

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

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.
 Files:
9424.tex