Percolation of Finite Clusters and Infinite Surfaces

Two related issues are explored for bond percolation on Z^d (with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point p_fin satisfies p_fin ≥ p_c, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spread out. It is not known whether d ≥ 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point p_surf satisfies p_surf ≥ p_fin.