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

We consider supercritical bond percolation in ${\mathbb{Z}}^d$ for $d\ge 3$. 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 $\mathbf{\ell }$, we prove sharp asymptotics for the probability that this cluster contains a vertex $x\in {\mathbb{Z}}^d$ that satisfies $x\cdot \mathbf{\ell }\ge \mathbf{u}$. For an axially aligned $\mathbf{\ell }$, we find this probability to be of the form $\kappa \exp \{-\zeta u\}(1+\mathrm{e}\mathrm{r}\mathrm{r})$ for $u\in \mathbb{N}$, where $\mathrm{\mid }\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{\mid }$ is at most $C\exp \{-c{u}^{1\mathrm{/}2}\}$; for general $\mathbf{\ell }$, the form of the asymptotic depends on whether $\mathbf{\ell }$ 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 $C\exp \{-c{u}^{1\mathrm{/}2}\}$.