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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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