93-3 Toom A.

More on Critical Phenomena in Growth Systems (78K, LateX) Jan 6, 93

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Abstract. This paper treats of interacting infinite or finite systems whose components' states are in the set \MB{\SET{0,1,2,\ldots}}. 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. Theorem 1 states conditions under which an infinite deterministic (\MB{\theta=0}) system is an `eroder', that is every initial condition with a finite set of positive components turns into `all zeroes' after a finite time. Theorems 2,3,4 are about probabilistic systems with initial states `all zeroes'. 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 the latter ways of behavior are given here, which are also necessary under natural assumptions.

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