Electronic Journal of Probability

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

Harry Kesten, Vladas Sidoravicius, and Yu Zhang

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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 $C_i \gt 0$ 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 $[-n,n]^2$.

Article information

Electron. J. Probab., Volume 6 (2001), paper no. 4, 27 pp.

Accepted: 12 February 2001
First available in Project Euclid: 19 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Percolation close-packing

This work is licensed under aCreative Commons Attribution 3.0 License.


Kesten, Harry; Sidoravicius, Vladas; Zhang, Yu. Percolation of Arbitrary words on the Close-Packed Graph of $\mathbb{Z}^2$. Electron. J. Probab. 6 (2001), paper no. 4, 27 pp. doi:10.1214/EJP.v6-77. https://projecteuclid.org/euclid.ejp/1461097634

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