zbMATH — the first resource for mathematics

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
Full Text: