Some results about ergodicity in shape for a crystal growth model

François Ezanno (Aix-Marseille Université)

Abstract


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.

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Pages: 1-20

Publication Date: March 7, 2013

DOI: 10.1214/EJP.v18-2177

References

  • Gates, D. J.; Westcott, M. Kinetics of polymer crystallization. I. Discrete and continuum models. Proc. Roy. Soc. London Ser. A 416 (1988), no. 1851, 443--461. MR0941479
  • Gates, D. J.; Westcott, M. Kinetics of polymer crystallization. II. Growth régimes. Proc. Roy. Soc. London Ser. A 416 (1988), no. 1851, 463--476. MR0941480
  • Andjel, E. D.; Menshikov, M. V.; Sisko, V. V. Positive recurrence of processes associated to crystal growth models. Ann. Appl. Probab. 16 (2006), no. 3, 1059--1085. MR2260057
  • Gates, D. J.; Westcott, M. Markov models of steady crystal growth. Ann. Appl. Probab. 3 (1993), no. 2, 339--355. MR1221155
  • D. M. Sadler. On the growth of two dimensional crystals: 2. Assessment of kinetic theories of crystallization of polymers. phPolymer, 28(9):1440--1455, 1987
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