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

TL;DR

This paper establishes precise asymptotic probabilities for finite clusters in supercritical bond percolation in high dimensions, revealing exponential decay rates and the influence of lattice directions.

Contribution

It provides sharp asymptotic formulas for cluster connection probabilities in supercritical percolation, including detailed error bounds and lattice condition effects.

Findings

Probability of large cluster vertices decays exponentially with distance.

Renewal points in clusters are abundant with fast-decaying tail probabilities.

Asymptotics depend on the alignment of the direction vector with the lattice.

Abstract

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 $C \exp \big\{ - c u^{1/2} \big\}$.