{"paper":{"title":"A Geometrical Structure for an Infinite Oriented Cluster and its Uniqueness","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xian-Yuan Wu, Yu Zhang","submitted_at":"2006-03-24T16:25:25Z","abstract_excerpt":"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\n $G$ 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603580","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}