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Exponential waiting time for filling a large interval in the symmetric simple exclusion process. (English) Zbl 0819.60095
The one-dimensional Markovian process with the nearest neighbor interaction is considered, i.e. the state space of the process is a collection of subsets in $$\mathbb{Z}$$. Probabilistic measure of the process is defined by means of the following condition. Let $$\mathbb{A} \subseteq \mathbb{Z}$$ be the random state at time $$t$$ and $$\mathbb{A}' \subseteq \mathbb{Z}$$ be a state at time $$t + \Delta$$. Then the conditional probabilities of the transitions $$\mathbb{A} \to \mathbb{A}'$$ are equal to each other if $$\mathbb{A}'$$ differs from $$\mathbb{A}$$ by the variation of the $$\mathbb{A}$$-neighbors. Namely, if both nearest neighbors $$x,y$$ are contained in $$\mathbb{A}$$, then $$x,y \in \mathbb{A}'$$; if $$x,y$$ are not contained in $$\mathbb{A}$$, then $$x,y \notin \mathbb{A}'$$ and if $$x \in \mathbb{A}, y \notin \mathbb{A}$$, then $$x \notin \mathbb{A}'$$, $$y \in \mathbb{A}'$$. In other cases the variation rate is equal to zero. Such random process is called a symmetric simple exclusion one. The extremal invariant measures for this process are the product measure $$(\nu_ \rho)^ \mathbb{Z}$$, $$\rho \in [0,1]$$, and $$\nu_ \rho$$ is such that the site $$x \in \mathbb{Z}$$ is occupied with the probability $$\rho$$. Let $$T_ N$$ be the first time that the sites $$\{1,2, \dots, N\}$$ get occupied, i.e. $$\{1,2, \dots, N\} \subset \mathbb{A}$$. The following statement is proved: There exist $$0 < \alpha' \leq \alpha_ N \leq \alpha'' < \infty$$ and positive $$A$$ and $$A'$$ such that $\sup_{t \geq 0} \bigl | \mathbb{P} \{\alpha_ N \rho^ N T_ N > t\} < e^{-t} \bigr | < A \rho^{A'N}.$

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics 60F10 Large deviations
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