One-point boundaries of ends of clusters in percolation in $\mathbb H^d$

TL;DR

This paper investigates the boundary properties of percolation clusters in hyperbolic space, establishing that below a certain critical probability, all clusters have boundary ends consisting of a single point.

Contribution

It introduces the concept of a critical percolation parameter in hyperbolic space and proves that below this threshold, clusters have one-point boundaries of ends.

Findings

Existence of a critical percolation parameter $p_0$ in hyperbolic space.

Below $p_0$, all clusters have one-point boundaries of ends.

Clusters do not have boundary points at infinity with positive probability.

Abstract

Consider Bernoulli bond percolation on a graph nicely embedded in hyperbolic space $\mathbb H^d$ in such a way that it admits a transitive action by isometries of $\mathbb H^d$. Let $p_0$ be the supremum of such percolation parameters that no point at infinity of $\mathbb H^d$ lies in the boundary of the cluster of a fixed vertex with positive probability. Then for any parameter $p < p_0$, a.s. every percolation cluster has only one-point boundaries of ends.