Essential enhancements revisited

Abstract: In 1991 Aizenman and Grimmett claimed that any `essential enhancement' of site or bond percolation on a lattice lowers the critical probability, an important result with many implications, such as strict inequalities between critical probabilities on suitable pairs of lattices. Their proof has two parts, one probabilistic and one combinatorial. In this paper we point out that a key combinatorial lemma, for which they provide only a figure as proof, is false. We prove an alternative form of the lemma, and thus the enhancement result, in the special cases of site percolation on the square, triangular and cubic lattices, and for bond percolation on $Z^d$, $d\ge 2$. The general case remains open, even for site percolation on $Z^d$, $d\ge 4$.