Some results about ergodicity in shape for a crystal growth model

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We study a crystal growth Markov model proposed by Gates and Westcott (Kinetics1, 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 (βᵢ, 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 AMS and 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.