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On critical phenomena in interacting growth systems. Part II: Bounded growth

Abstract

This paper completes the classification of some infinite and finite growth systems which was started in Part I. Components whose states are integer numbers interact in a local deterministic way, in addition to which every component's state grows by a positive integerk with a probability εk(1-ε) at every moment of the discrete time. Proposition 1 says that in the infinite system which starts from the state “all zeros”, percentages of elements whose states exceed a given valuek≥0 never exceed (Cε)k, whereC=const. Proposition 2 refers to finite systems. It states that the same inequalities hold during a time which depends exponentially on the system size.

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References

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    Andrei Toom, Stable and attractive trajectories in multicomponent systems inMulticomponent Random Systems, R. Dobrushin and Ya. Sinai, eds. (Marcel Dekker, New York, 1980) pp. 549–576.

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    A. Toom, N. Vasilyev, O. Stavskaya, L. Mityushin, G. Kurdryumov, and S. Pirogov, Discrete local Markov systems, inStochastic Cellular Systems: Ergodicity, Memory, Morphogenesis, R. Dobrushin, V. Kryukov, and A. Toom, eds. (Manchester University Press, 1990).

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    Andrei Toom, On critical phenomena in interacting growth systems. Part I: General,J. Stat. Phys., this issue.

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Toom, A. On critical phenomena in interacting growth systems. Part II: Bounded growth. J Stat Phys 74, 111–130 (1994). https://doi.org/10.1007/BF02186809

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Key Words

  • Random process
  • local interaction
  • critical phenomena growth
  • combinatorics
  • contour method
  • graph theory