Some results about ergodicity in shape for a crystal growth model

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.