Limit Theorems for the Size of the Clusters of Dependent Percolation Process on Z²

Limit theorems such as central limit theorems, large deviations and large number laws play an important role in probability theory. In this paper, we consider dependent percolation on the planar lattice \mathbbZ^2. For this model, we not only prove the existence and uniqueness of the infinite cluster but also prove the central limit theorems respectively for the size of biggest cluster and the size of the cluster at the origin in the lattice boxes