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Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice
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Harry Kesten, Cornell University
Vladas Sidoravicius, IMPA
Yu Zhang, University of Colorado
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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 $0 < be < 1$).
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Full text: PDF
Pages: 1-75
Published on: July 7, 1998
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Electronic Journal of Probability. ISSN: 1083-6489 |
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