The supercritical phase of percolation is well behaved

Source abstract

Abstract We prove a general result concerning the critical probabilities of subsets of a lattice L. It is a consequence of this result that the critical probability of a percolation process on L equals the limit of the critical probability of a slice of L as the thickness of the slice tends to infinity. This verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation.