Stationary measures for an exclusion process on one-dimensional lattices with infinitely many hopping sites (Q1820518)

Stationary measures for the following exclusion process on one- dimensional lattices with discrete time parameter are considered: As time increases from t to \(t+1\), each particle which locates at the site i moves to the site i-1 independently with probability \(\alpha\) \((0<\alpha <1/2)\) iff the site i-1 is unoccupied. It is known that the stationary measures for the process are the convex combinations of a family of nearest neighbor Gibbs states (renewal measures) \(\{\pi_{\gamma}:\) \(0<\gamma <\infty \}\) and trivial Dirac measures \(\{\pi_ 0,\pi_{\infty},\Theta_ n,n\in {\mathbb{Z}}\}\). Concerning the stochastic property of a tagged particle under the time evolution with respect to the stationary state \(\pi_{\gamma}\), it is shown that the position of tagged particle \(\{\) r(t), \(t=0,1,2,...\}\) has independent increments, i.e., \(\{\) r(t)-r(t-1)\(\}\) is a Bernoulli sequence, and so a central limit theorem holds. By applying a simple limiting procedure to our process we can obtain a usual simple exclusion process with continuous time parameter.