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The contact process in a random environment. (English) Zbl 0741.60097
This paper deals with one-dimensional contact process in an i.i.d. environment (CPRE). Let \(\xi_ t\subset\mathbb{Z}\) be the state of the process at time \(t\). Then, the dynamics are formulated as follows: a) Particles are born at unoccupied sites \(x\) at rate \(|\xi_ t\cap\{x- 1, x+1\}|\). b) A particle at \(x\) dies out at rate \(\delta_ x\), where \(\delta_ x\) (\(x\in\mathbb{Z}\)) are i.i.d. having values \(\Delta\) and \(\delta\) with probabilities \(\mathbb{P}[\delta_ x=\Delta]=p\) and \(\mathbb{P}[\delta_ x=\delta]=1-p\), respectively. The authors prove that if \(p<1\), then for sufficiently small \(\delta\), depending on \(\Delta\) and \(p\), the CPRE survives for a.e. environment. This describes the first phase transition. The second one means that there is a region in the surviving area in which the CPRE grows linearly. Some open problems for further study are also mentioned.

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