Random context-free grammars: supercritical case with a nonzero extinction probability

Random grammars were introduced in information science; however, investigating them with the help of methods of statistical physics, such as investigation of the thermodynamic limit, methods of cluster expansions, etc., started recently; see V. A. Malyshev [Russian Math. Surveys, 53 (1998), pp. 345–370]. For context-free grammars in a supercritical case with zero extinction probability (where symbols do not die) in [F. I. Karpelevich, V. A. Malyshev, A. I. Petrov, S. A. Pirogov, and A. N. Rybko, Context Free Evolution of Words, Rapport de Recherche No. 4413, INRIA, Rocquencourt, France, 2002] the large time behavior was studied, the existence of different limit measures was proved, and the connection between them was considered. In this paper we expand the main results of Karpelevich et al. to the supercritical case with nonzero extinction probability.