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The survival of one-dimensional contact processes in random environments. (English) Zbl 0754.60126
Interacting particle systems in random environments have recently obtained a lot of attention. This paper considers the inhomogeneous contact process on $$Z^ 1$$ with recovery rate $$\delta(k)$$ at site $$k$$ and infection rates $$\lambda(k)$$ and $$\rho(k)$$ at site $$k$$ due to the presence of infected neighbors at $$k-1$$ and $$k+1$$, respectively. A special case of the main result is the following: Suppose that the environment is chosen in such a way that the $$\delta(k)$$’s, $$\lambda(k)$$’s and $$\rho(k)$$’s are all mutually independent, with the $$\delta(k)$$’s having a common distribution, and the $$\lambda(k)$$’s and $$\rho(k)$$’s having a common distribution. Then the process survives if $\mathbb{E}{\delta(\lambda+\rho+\delta)\over \lambda\rho}<1.$ If the environment is deterministic and periodic with period $$p$$, then the process survives if $\prod^ p_{k=1}{\delta(k)[\lambda(k)+\rho(k- 1)+\delta(k)]\over\lambda(k)\rho(k- 1)}<1\quad\text{and}\quad \prod^ p_{k=1}{\delta(k-1)[\lambda(k)+\rho(k-1)+\delta(k- 1)]\over\lambda(k)\rho( k-1)}<1.$

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