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May 2008 A geometrical structure for an infinite oriented cluster and its uniqueness
Xian-Yuan Wu, Yu Zhang
Ann. Probab. 36(3): 862-875 (May 2008). DOI: 10.1214/07-AOP339


We consider the supercritical oriented percolation model. Let ${\mathscr {K}}$ be all the percolation points. For each $u\in{ \mathscr {K}}$, we write γu as its rightmost 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 ${\mathscr {K}}$ 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.


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Xian-Yuan Wu. Yu Zhang. "A geometrical structure for an infinite oriented cluster and its uniqueness." Ann. Probab. 36 (3) 862 - 875, May 2008.


Published: May 2008
First available in Project Euclid: 9 April 2008

zbMATH: 1139.60345
MathSciNet: MR2408576
Digital Object Identifier: 10.1214/07-AOP339

Primary: 60K35 , 82B43

Keywords: Oriented percolation , topological ends of graph , uniqueness of infinite cluster

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 3 • May 2008
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