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1998 Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice
Harry Kesten, Vladas Sidoravicius, Yu Zhang
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Electron. J. Probab. 3: 1-75 (1998). DOI: 10.1214/EJP.v3-32

Abstract

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 $0 <\beta < 1$).

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Harry Kesten. Vladas Sidoravicius. Yu Zhang. "Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice." Electron. J. Probab. 3 1 - 75, 1998. https://doi.org/10.1214/EJP.v3-32

Information

Accepted: 7 July 1998; Published: 1998
First available in Project Euclid: 29 January 2016

zbMATH: 0908.60082
MathSciNet: MR1637089
Digital Object Identifier: 10.1214/EJP.v3-32

Subjects:
Primary: 60K35

Keywords: percolation , Triangular lattice

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Vol.3 • 1998
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