Toom A.
A Critical Phenomenon in Growth Systems
(73K, LateX)
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}.