  We study a crystal growth Markov model proposed by Gates and Westcott
(\cite{Kinetics1}, \cite{Kinetics2}). This is an aggregation process where
particles are packed in a square lattice accordingly to prescribed deposition
rates. This model is parametrized by three values $(\beta_i, i=0,1,2)$
corresponding to depositions on three different types of sites. The main
problem is to determine, for the shape of the crystal, when recurrence and when
ergodicity do occur. In \cite{AMS} and \cite{MarkovModels} sufficient
conditions are given both for ergodicity and transience. We establish some
improved conditions and give a precise description of the asymptotic behavior
in a special case.
