Percolation of finite clusters and shielded paths

Abstract

In independent bond percolation on $\mathbb{Z}^d$ 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 cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for $d \geq 19$, 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 $d \geq 11$. We improve this result by showing that for $d \geq 10$ 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 numerical values of $p_c$, this bound can be reduced to $d \geq 7$. Our methods are elementary and do not require the triangle condition.