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 Electronic Journal of Probability > Vol. 3 (1998) > Paper 10 open journal systems 


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

Harry Kesten, Cornell University
Vladas Sidoravicius, IMPA
Yu Zhang, University of Colorado


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 $0 < be < 1$).


Full text: PDF

Pages: 1-75

Published on: July 7, 1998


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Electronic Journal of Probability. ISSN: 1083-6489