A Reversible Nearest Particle System on the Homogeneous Tree

Amber L. Puha ()
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Amber L. Puha: University of California

Journal of Theoretical Probability, 1999, vol. 12, issue 1, 217-254

Abstract: Abstract We introduce a modified contact process on the homogeneous tree. The modification is to the death rate: an occupied site becomes vacant at rate one if the number of occupied id neighbors is at most one. This modification leads to a growth model which is reversible, off the 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 this paper is that there is exactly one phase transition on the binary tree. Furthermore, the value of the birth parameter at which the phase transition occurs is explicitly computed In particulars survival and complete convergence hold if the birth parameter exceeds 1/4. Otherwise, the expected extinction time is finite.

Keywords: Trees; growth models; phase transition; reversible; flows (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1023/A:1021709013569

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