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 $d \geq 3$. 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 $\bf\ell$, we prove sharp asymptotics for the probability that this cluster contains a vertex $x \in \mathbb{Z}^d$ that satisfies $x \cdot \bf\ell \geq u$. For an axially aligned $\bf\ell$, we find this probability to be of the form $\kappa \exp \{ - \zeta u \}(1+ {\rm err})$ for $u \in \mathbb{N}$, where $\vert {\rm err} \vert$ is at most $C \exp \{ - c u^{1/2} \big\}$; for general $\bf\ell$, the form of the asymptotic depends on whether $\bf\ell$ 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 $C \exp \big\{ - c u^{1/2} \big\}$.