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Finite nearest particle systems. (English) Zbl 0557.60087
The model considered here generalizes the Harris contact process (state space: the finite subsets of Z) in the following way: the death rate of any particle is again identically one, the birth rate at a site is a symmetric function \(\beta\) (l,r) of its distance to the nearest occupied site to the left and right of it, which satisfies an irreducibility and non-explosion condition. The problem is to find criteria for extinction (which means extinction a.s.) and survival (which means survival with positive probability) in terms of \(b_ n:=\sum_{l+r=n}\beta (l,r)\), \(n\geq 2\). The result is contained in
Theorem 1.7. (1) If \(b_ n\leq 1\) for all n, the process dies out.
(2) To each \(b>2\) there is a nearest particle process with \(b_ n=b\) for all n, which survives.
(3) If \(b_ n\geq 4\) for all n, the process survives.
The most interesting part is (3), for the proof of which a particular comparison between the given process and a contact process has to be constructed.
Reviewer: H.Rost

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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