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Some results about ergodicity in shape for a crystal growth model. (English) Zbl 1288.82057
Summary: We study a crystal growth Markov model proposed by Gates and Westcott. 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. Sufficient conditions are known both for ergodicity and transience. We establish some improved conditions and give a precise description of the asymptotic behavior in a special case.
##### MSC:
 82D25 Statistical mechanics of crystals 60J27 Continuous-time Markov processes on discrete state spaces 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
##### Keywords:
Markov chain; random deposition; positive recurrence
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