Abstract.
Let ? d be a homogeneous tree in which every vertex has d + 1 neighbours, where d≥ 2. The contact process on such a tree is known to have three distinct phases. We consider the process on a finite subtree, namely the rooted tree of depth h and branching factor d, and relate the behaviour of the process on the infinite tree to its behaviour on the finite tree for large h. In the phase of strong survival, we show that with probability ɛ independent of h, the process on the subtree starting from a single infection survives for a time which is doubly exponential in h and almost exponential in the number of vertices of the finite tree. In the phase of weak survival on the infinite tree, the survival time on the finite tree is approximately linear in h. In the phase of no survival, the survival time on the finite tree is linear in h if one starts with all vertices initially infected, and bounded by a random variable (independent of h) with an exponential tail if one starts from a single infection.
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Received: 14 April 2000 / Revised version: 6 January 2001 / Published online: 9 October 2001
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Stacey, A. The contact process on finite homogeneous trees. Probab Theory Relat Fields 121, 551–576 (2001). https://doi.org/10.1007/s004400100149
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DOI: https://doi.org/10.1007/s004400100149
Keywords
- Survival Time
- Rooted Tree
- Distinct Phasis
- Strong Survival
- Homogeneous Tree