Stochastic Growth Models

  • R. Durrett
  • R. H. Schonmann
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 8)


This paper is based on a talk given by the first author at the I.M.A. in February, 1986 but incorporates improvements discovered during six later repiti-tions. The second authour should not be held responsible for the style of presentation of the results but should be given credit for discovering the results independently in the Fall of 1985. The discussion below is equal to the talk with most of the details of the proofs filled in, but we have tried to preserve the informal style of the talk and concentrate on the “main ideas” rather than giving complete details of the proofs. If we forget about definitions then the results can be summed up in a few words “Everything Durrett and Griffeath (1983) proved for one-dimensional nearest neighbor additive groth models is true for the corresponding class of finite range models, i.e., those which can be constructed from a percolation structure.”


Cellular Automaton Percolation Process Contact Process Open Bond Bond Percolation 
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  1. Amati, D., et al (1976) Expanding disc as dynamic vacuum instability in Reggeon field theory. Nucl. Phys. B 114, 483–504.CrossRefGoogle Scholar
  2. Brower, R.C., Furman, M.A., and Moshe, M, (1978) Critical exponents for the Reggeon quantum spin model. Phys. Lett B. 76, 213–219.CrossRefGoogle Scholar
  3. Cardy, J.L and Sugar, R.L. (1980) Directed percolation and Reggeon field theory. J. Phys. A 13, L423–L427.MathSciNetCrossRefGoogle Scholar
  4. Dhar, D. and Barma (1981) Monte Carlo simulation of directed percolation on the square lattice. J. Phys. C, L1–6.Google Scholar
  5. Domany, E. and Kinzel, W. (1981) Directed percolation in two dimensions: numerical analysis and an exact solution Phys. Rev. lett. 47, 1238–1241.MathSciNetCrossRefGoogle Scholar
  6. Durrett, R. (1980) On the growth of one dimensional contact processes. Ann. Prob. 8, 890–907.MathSciNetMATHCrossRefGoogle Scholar
  7. Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.MathSciNetMATHCrossRefGoogle Scholar
  8. Durrett, R., and Griffeath, D. (1982) Contact processes in several dimensions, Z. fur Wahr. 59, 535–552.MathSciNetMATHCrossRefGoogle Scholar
  9. Durrett, R. and Griffeath, D. (1983) Supercritical contact processes on Z. Ann. Prob. 11, 1–15.MathSciNetMATHCrossRefGoogle Scholar
  10. Grassberger, P. and de la Torre, A. (1979) Reggeon field theory (Shlogl’s first model) on a lattice: Monte Carlo calculations of critical behavior. Ann. Phys. 122, 373–396.CrossRefGoogle Scholar
  11. Griffeath, D. (1979) Additive and Cancellative Interacting Particle Systems Springer Lecture Notes in Math, Vol. 724.MATHGoogle Scholar
  12. Griffeath, D. (1981) The basic contact process. Stoch. Pro. Appl. 11, 151–168.MathSciNetMATHCrossRefGoogle Scholar
  13. Harris, T.E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969–988.MATHCrossRefGoogle Scholar
  14. Harris, T.E. (1978) Additive set valued processes and graphical methods. Ann. Prob. 6, 355–378.MATHCrossRefGoogle Scholar
  15. Kertesz, J. and Vicsek, T. (1980) Oriented bond percolation. J. Phys. C. 13, L343–348.CrossRefGoogle Scholar
  16. Kinzel, W. and Yeomans, J. (1981) Directed percolation: a finite size renor-malization approach. J. Phys. A14, L163–168.Google Scholar
  17. Kinzel, W. (1985) Phase transitions of cellular automata. Z. Phys. B. 58, 229–244.MathSciNetMATHCrossRefGoogle Scholar
  18. Liggett, T. Interacting Particle Systems. Springer-Verlag, New York.Google Scholar
  19. Schonmann, R.H. (1985) Metastability for the contact process. J. Stat. Phys. 41, 445–464.MathSciNetMATHCrossRefGoogle Scholar
  20. Schonmann, R.H. (1986a) A new proof of the complete convergence theorem for contact processes in several dimensions with large infection parameter. Ann. Prob., to appear.Google Scholar
  21. Schonmann, R.H. (1986b) The asymmetric contact process. J. Stat. Phys., to appear.Google Scholar
  22. Wolfram, S. (1983) Statistical mechanics of cellular automata. Rev. Mod. Phys. 55, 601–650.MathSciNetMATHCrossRefGoogle Scholar
  23. Wolfram, S. (1984) Universality and complexity in cellular automata. Phsica 10 D, 1–35.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1987

Authors and Affiliations

  • R. Durrett
    • 1
  • R. H. Schonmann
    • 2
    • 3
  1. 1.CornellUSA
  2. 2.Sao PauloBrazil
  3. 3.RutgersUSA

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