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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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