93-2 Toom A.

A Critical Phenomenon in Growth Systems (73K, LateX) Jan 6, 93

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Abstract. The paper treats of interacting infinite or finite systems whose components' states are in the set \mb{\SET{0,1,2,3,\ldots}}. All components' initial states are zeroes. Components interact with each other in a local deterministic way, in addition to which every component's state grows by one with a constant probability \mb{\theta} at every moment of the discrete time. Our main question about infinite systems is whether the average value of components tends to infinity or remains bounded as \mb{t\to\infty}. The analogous question about finite systems is how long the system's average remains less than a constant: this time may be bounded or tend to infinity as the system' size tends to infinity. Both in the infinite and finite cases sufficient conditions for both ways of behavior are given here. It is shown that the different ways of behavior may occur with one and the same deterministic interaction depending on the value of \mb{\theta}.

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