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 \ - u \ (1+ err) for u N, where err is at most C \ - c u^{1/2 \}; 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 \ - c u^{1/2 \}.