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

   
@article{EJP32,
	author = {Harry Kesten and Vladas Sidoravicius and Yu Zhang},
	title = {Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {3},
	year = {1998},
	keywords = {Percolation, Triangular lattice},
	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 $0 <\beta < 1$).},
	pages = {no. 10, 1-75},
	issn = {1083-6489},
	doi = {10.1214/EJP.v3-32},    
        url = {http://ejp.ejpecp.org/article/view/32}}