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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