A geometrical structure for an infinite oriented cluster and its uniqueness

Abstract

We consider the supercritical oriented percolation model. Let ${\mathscr {K}}$ be all the percolation points. For each $u\in{ \mathscr {K}}$, we write γ_u as its rightmost path. Let G=⋃_uγ_u. In this paper, we show that G is a single tree with only one topological end. We also present a relationship between ${\mathscr {K}}$ and G and construct a bijection between ${\mathscr {K}}$ and ℤ using the preorder traversal algorithm. Through applications of this fundamental graph property, we show the uniqueness of an infinite oriented cluster by ignoring finite vertices.