A reversible nearest particle system on the homogeneous tree
It is introduced a modified contact process on the homogeneous (connected) tree in which each vertex has \(d+1\) neighbors [for definitions and some results see \textit{T. M. Liggett}, ``Interacting particle systems'' (1985; Zbl 0559.60078), Ann. Probab. 24, No. 4, 1675-1710 (1996; Zbl 0871.60087) and \textit{D. Chen}, Acta Math. Sci. 14, No. 3, 348-353 (1994; Zbl 0814.60101)]. The modification is to the death rate: an occupied site becomes vacant at rate one if the number of its occupied neighbors is at most one. This modification leads to a growth model that is reversible, off empty set, provided the initial set of occupied sites is connected. Reversibility admits tools for studying the survival properties of the system not available in a nonreversible situation. Four potential phases are considered: extinction, weak survival, strong survival, and complete convergence. The main result of the paper is that there is exactly one phase transition on the binary tree. The value of the birth parameter at which the phase transitions occur is explicitly computed. In particular, survival and complete convergence hold if the birth parameter exceeds \(1/4\). Otherwise, the expected extinction time is finite.
scientific article; zbMATH DE number 1275655