Asymptotic behavior of a tagged particle in the exclusion process on parallel lattices

We investigate a tagged particle in the exclusion processes on {1, ..., N }×ℤ d , with different densities in different levels { k }×ℤ d , ∀ k . Ignoring the level the tagged particle lying in, we only concern its position in ℤ d , denoted by X t . Note that the whole space is not homogeneous. We define the environment process viewed from the tagged particle, of which X t can be expressed as a functional. It is called the tagged particle process. We show the ergodicity of the tagged particle process, then prove the strong law of large numbers. Furthermore, we show the central limit theorem of X t provided the zero-mean condition.