On the uniqueness of the infinite occupied cluster in dependent two- dimensional site percolation
This paper considers dependent site percolation on the two-dimensional square lattice. The underlying probability measure is assumed to be invariant and ergodic under either horizontal or vertical translation separately, and invariant under axis reflections. If the FKG condition is satisfied and if percolation occurs, then the infinite occupied cluster is unique with probability 1. The proof uses Harris' technique for showing uniqueness of the infinite percolation cluster, i.e., demonstrating the existence of circuits of occupied sites surrounding arbitrarily large boxes. A multiple ergodic lemma similar to a result of Furstenburg provides key inequalities for the probabilities that certain chains of occupied sites avoid the boxes. Symmetries and the FKG inequality then build the chains into closed circuits around the boxes. It is not known whether the result generalizes to higher dimensions than two, nor whether the hypotheses of the theorem can be weakened.
scientific article; zbMATH DE number 4076275