A Geometrical Structure for an Infinite Oriented Cluster and its Uniqueness

Abstract

We consider the supercritical oriented percolation model. Let ${\fK}$ be all the percolation points. For each $u\in {\fK}$, we write $\gamma_u$ as its right-most path. Let $G=\cup_u \gamma_u$. In this paper, we show that $G$ is a single tree with only one topological end. We also present a relationship between ${\fK}$ and $G$ and construct a bijection between ${\fK}$ 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.