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Percolation of Arbitrary words on the Close-Packed Graph of $\mathbb{Z}^2$

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Percolation of Arbitrary words on the Close-Packed Graph of $\mathbb{Z}^2$
2.	Creator	Author's name, affiliation, country	Harry Kesten; Cornell University
2.	Creator	Author's name, affiliation, country	Vladas Sidoravicius; IMPA
2.	Creator	Author's name, affiliation, country	Yu Zhang; University of Colorado
3.	Subject	Discipline(s)	Mathematics
3.	Subject	Keyword(s)	Percolation, close-packing
3.	Subject	Subject classification	Primary. 60K35
4.	Description	Abstract	Let ${\Bbb Z}^2_{cp}$ be the close-packed graph of $\Bbb Z^2$, that is, the graph obtained by adding to each face of $\Bbb Z^2$ its diagonal edges. We consider site percolation on $\Bbb Z^2_{cp}$, namely, for each $v$ we choose $X(v) = 1$ or 0 with probability $p$ or $1-p$, respectively, independently for all vertices $v$ of $\Bbb Z^2_{cp}$. We say that a word $(\xi_1, \xi_2,\dots) \in \{0,1\}^{\Bbb N}$ is seen in the percolation configuration if there exists a selfavoiding path $(v_1, v_2, \dots)$ on $\Bbb Z^2_{cp}$ with $X(v_i) = \xi_i, i \ge 1$. $p_c(\Bbb Z^2, \text{site})$ denotes the critical probability for site-percolation on $\Bbb Z^2$. We prove that for each fixed $p \in \big (1- p_c(\Bbb Z^2, \text{site}), p_c(\Bbb Z^2, \text{site})\big )$, with probability 1 all words are seen. We also show that for some constants $C_i > 0$ there is a probability of at least $C_1$ that all words of length $C_0n^2$ are seen along a path which starts at a neighbor of the origin and is contained in the square $[-n,n]^2$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2001-02-12
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/77
10.	Identifier	Digital Object Identifier	10.1214/EJP.v6-77
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 6
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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