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The periodic threshold contact process. (English) Zbl 0747.60097
Random walks, Brownian motion, and interacting particle systems, Festschr. in Honor of Frank Spitzer, Prog. Probab. 28, 339-358 (1991).
[For the entire collection see Zbl 0733.00027.]
The periodic threshold contact process is the Markov process on $$\{0,1\}^ \mathbb{Z}$$ with the following transitions: \begin{aligned} 1 &\to 0 \text{ at site $$k$$ at rate } 1,\\ 0 &\to 1 \text{ at site $$k$$ at rate $$\lambda$$, if $$k$$ is odd and there is a 1 at either $$k-1$$ or at } k+1,\\ 0 &\to 1 \text{ at site $$k$$ at rate $$\mu$$, if $$k$$ is even and there is a 1 at either $$k-1$$ or at } k+1.\\ \end{aligned} The process is said to survive if there is an invariant measure other than the point mass on the empty set. It is proved that the process dies out if $$\lambda+\mu+2>4\lambda\mu$$. A sufficient condition is given for its survival, and this condition is satisfied at $$(\lambda,\mu)=(2.17,2.18),(2.00,2.37),(1.50,3.62)$$, for example.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
contact process; invariant measure