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Persistent survival of one-dimensional contact processes in random environments. (English) Zbl 0863.60098
The authors consider a one-dimensional contact process in a random environment. It is defined as a Markov process with state space $$\{0,1\}^{\mathbb{Z}}$$ with transitions $$1\to 0$$ at site $$x$$ with rate $$\delta(x)$$ and $$0\to 1$$ at site $$x$$ with rate $$\varepsilon[\eta_t(x-1)+\eta_t(x+1)]$$, where $$\eta_t$$ is the state of the process. The focus of the article is on the event that the process $$\eta_t$$ starting from the point $$\eta_0^{(j)}(x)=\begin{cases} 1, &x=j,\\ 0, &x\neq j,\end{cases}$$ is not identically zero for every $$t>0$$ (i.e. survives). In particular, it is proved that if $$u \mathbb{P}(-\log\delta(x)>u)$$ tends to $$+\infty$$ as $$u\to +\infty$$, then $$\eta_t$$ survives for every $$\varepsilon >0$$. The discrete and continuous time processes are considered.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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##### References:
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