A Geometrical Structure for an Infinite Oriented Cluster and Its Uniqueness

We consider the supercritical oriented percolation model. Let K be all the percolation points. For each u ∈ K, we write γu as its right-most path. Let G = ∪uγu. In this paper, we show that G is a single tree with only one topological end. We also present a relationship between K and G and construct a bijection between 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.