Sharp asymptotics for finite point-to-plane connections in supercritical bond percolation in dimension at least three

We consider supercritical bond percolation in Zᵈ for d ≥ 3. The origin lies in a finite open cluster with positive probability, and, when it does, the diameter of this cluster has an exponentially decaying tail. For each unit vector , we prove sharp asymptotics for the probability that this cluster contains a vertex x ∈ Zᵈ that satisfies x · ≥ u. For an axially aligned , we find this probability to be of the form κ exp \ - ζ u \(1+ err) for u ∈ N, where err is at most C exp \ - c u1/2 \; for general , the form of the asymptotic depends on whether satisfies a natural lattice condition. To obtain these results, we prove that renewal points in long clusters are abundant, with a renewal block length whose tail is shown to decay as fast as C exp \ - c u1/2 \.