an:00740322
Zbl 0819.60095
Ferrari, P. A.; Galves, A.; Liggett, T. M.
Exponential waiting time for filling a large interval in the symmetric simple exclusion process
EN
Ann. Inst. Henri Poincar??, Probab. Stat. 31, No. 1, 155-175 (1995).
00025288
1995
j
60K35 82C22 60F10
symmetric exclusion process; occurrence time of a rare event; large deviations
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
\[
\sup_{t \geq 0} \bigl | \mathbb{P} \{\alpha_ N \rho^ N T_ N > t\} < e^{-t} \bigr | < A \rho^{A'N}.
\]
Y.P.Virchenko (Khar'kov)