Electronic Journal of Probability

Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice

Harry Kesten, Vladas Sidoravicius, and Yu Zhang

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We consider critical site percolation on the triangular lattice, that is, we choose $X(v) = 0$ or 1 with probability 1/2 each, independently for all vertices $v$ of the triangular lattice. 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 the triangular lattice with $X(v_i) = \xi_i, i \ge 1$. We prove that with probability 1 "almost all" words, as well as all periodic words, except the two words $(1,1,1, \dots)$ and $(0,0,0,\dots)$, are seen. "Almost all" words here means almost all with respect to the measure $\mu_\beta$ under which the $\xi_i$ are i.i.d. with $\mu_\beta {\xi_i = 0}=1 - \mu_\beta {\xi_i = 1} = \beta$ (for an arbitrary $0 <\beta < 1$).

Article information

Electron. J. Probab., Volume 3 (1998), paper no. 10, 75 pp.

Accepted: 7 July 1998
First available in Project Euclid: 29 January 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 Triangular lattice

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Kesten, Harry; Sidoravicius, Vladas; Zhang, Yu. Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice. Electron. J. Probab. 3 (1998), paper no. 10, 75 pp. doi:10.1214/EJP.v3-32. https://projecteuclid.org/euclid.ejp/1454101770

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