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Analysis of a stochastic model of the growth of trees. (English) Zbl 0871.60045
Summary: On a semi-infinite tree (that is, a tree with a “root”), only those configurations that are subtrees with the same root, called continuous configurations, are considered. A Markov process with discrete time and with phase space formed by the continuous configurations is studied. In this case the limit distribution of a Markov chain corresponds to the limit “Gibbs” distribution on the set of continuous configurations. A “phase transition” of these measures is described in dependence on the ratio of the birth and death probabilities; namely, it is proved that if this ratio exceeds a certain critical value, then the limit state of the system is the \(\delta\)-measure concentrated on the whole tree, and otherwise there exists a limit state concentrated on a set of finite subtrees of the tree.
60J05 Discrete-time Markov processes on general state spaces
05C05 Trees
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
90B15 Stochastic network models in operations research