Journal of Statistical Physics

, Volume 74, Issue 1–2, pp 111–130 | Cite as

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

  • Andrei Toom
Articles

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 ε k (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.

Key Words

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

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References

  1. 1.
    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.Google Scholar
  2. 2.
    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).Google Scholar
  3. 3.
    Andrei Toom, On critical phenomena in interacting growth systems. Part I: General,J. Stat. Phys., this issue.Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Andrei Toom
    • 1
  1. 1.Incarnate Word CollegeSan Antonio

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