Percolation of Finite Clusters and Shielded Paths

In independent bond percolation on Z^d with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for d≥ 19 , the set of such p contains values strictly larger than the percolation threshold p_c. With the work of Fitzner-van der Hofstad, this has been reduced to d≥ 11. We improve this result by showing that for d≥ 10 and some p> p_c, there are infinite paths consisting of “shielded” vertices—vertices all whose adjacent edges are closed—which must be in the complement of the infinite cluster. Using values of p_c obtained from computer simulations, this bound can be reduced to d≥ 7. Our methods are elementary and do not require the triangle condition.