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Extinction of contact and percolation processes in a random environment. (English) Zbl 0814.60098
Summary: We consider the (inhomogeneous) percolation process on $$\mathbb{Z}^ d \times \mathbb{R}$$ defined as follows: Along each vertical line $$\{x\} \times \mathbb{R}$$ we put cuts at times given by a Poisson point process with intensity $$\delta (x)$$, and between each pair of adjacent vertical lines $$\{x\} \times \mathbb{R}$$ and $$\{y\} \times \mathbb{R}$$ we place bridges at times given by a Poisson point process with intensity $$\lambda (x,y)$$. We say that $$(x,t)$$ and $$(y,s)$$ are connected (or in the same cluster) if there is a path from $$(x,t)$$ to $$(y,s)$$ made out of uncut segments of vertical lines and bridges.
If we consider only oriented percolation, we have the graphical representation of the (inhomogeneous) $$d$$-dimensional contact process. We consider these percolation and contact processes in a random environment by taking $$\delta = \{\delta (x);\;x \in \mathbb{Z}^ d\}$$ and $$\lambda = \{\lambda (x,y);\;x,y \in \mathbb{Z}^ d$$, $$\| x - y \|_ 2 = 1\}$$ to be independent families of independent identically distributed strictly positive random variables; we use $$\delta$$ and $$\lambda$$ for representative random variables. We prove extinction (i.e., no percolation) of these percolation and contact processes, for almost every $$\delta$$ and $$\lambda$$, if $$\delta$$ and $$\lambda$$ satisfy $$\mathbb{E} \{(\log (1 + \lambda))^ \beta\} < \infty$$ and $$\mathbb{E} \{(\log (1 + 1/ \delta))^ \beta\} < \infty$$ for some $$\beta > 2d^ 2(1 + \sqrt{1 + 1/d}+ 1/2d)$$, and if $$\mathbb{E} \{( \log (1 + \lambda/ \delta))^ \beta\}$$ is sufficiently small.

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