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 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.