We consider dependent site percolation on the two-dimensional square lattice, the underlying probability measure being invariant and ergodic under each of the translations and invariant under axis reflections. If this measure satisfies the FKG condition and if percolation occurs, then we show that the infinite occupied cluster is unique with probability 1, and that all vacant star-clusters are finite.
A. Gandolfi. M. Keane. L. Russo. "On the Uniqueness of the Infinite Occupied Cluster in Dependent Two- Dimensional Site Percolation." Ann. Probab. 16 (3) 1147 - 1157, July, 1988. https://doi.org/10.1214/aop/1176991681