In independent bond percolation on $${\mathbb {Z}}^d$$ 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 $$d \ge 19$$ d ≥ 19 , the set of such p contains values strictly larger than the percolation threshold $$p_c$$ p c . With the work of Fitzner-van der Hofstad, this has been reduced to $$d \ge 11$$ d ≥ 11 . We improve this result by showing that for $$d \ge 10$$ d ≥ 10 and some $$p>p_c$$ 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 $$p_c$$ p c obtained from computer simulations, this bound can be reduced to $$d \ge 7$$ d ≥ 7 . Our methods are elementary and do not require the triangle condition.

中文翻译：

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有限簇和屏蔽路径的渗透

在具有参数 p 的 $${\mathbb {Z}}^d$$ Z d 上的独立键渗流中，如果删除无限簇的顶点（和事件边），对于 p 的值，剩余的图是否包含一个无限连通分量？Grimmett-Holroyd-Kozma 使用三角形条件表明，对于 $$d \ge 19$$ d ≥ 19 ，此类 p 的集合包含严格大于渗透阈值 $$p_c$$ pc 的值。通过 Fitzner-van der Hofstad 的工作，这已经减少到 $$d \ge 11$$ d ≥ 11 。我们通过证明 $$d \ge 10$$ d ≥ 10 和一些 $$p>p_c$$ p > pc 来改进这个结果，存在由“屏蔽”顶点组成的无限路径——所有相邻边都是闭合的顶点——它必须在无限簇的补集中。使用从计算机模拟获得的 $$p_c$$ pc 值，这个界限可以减少到 $$d \ge 7$$ d ≥ 7 。我们的方法是基本的，不需要三角形条件。