Percolation of Arbitrary words on the Close-Packed Graph of Z2
Harry Kesten, Cornell University
Vladas Sidoravicius, IMPA
Yu Zhang, University of Colorado
Abstract
Let $Bbb Z^2_{cp}$ be the close-packed graph of $Bbb Z^2$,
that is, the graph obtained by adding to each face of
$Bbb Z^2$ its diagonal edges. We consider site percolation
on $Bbb Z^2_{cp}$, namely, for each $v$ we choose $X(v) = 1$ or 0 with
probability $p$ or $1-p$, respectively, independently
for all vertices $v$ of $Bbb Z^2_{cp}$. 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 $Bbb Z^2_{cp}$ with $X(v_i) = xi_i, i ge
1$. $p_c(Bbb Z^2, text{site})$ denotes the critical
probability for site-percolation on $Bbb Z^2$.
We prove that for each fixed $p in big (1- p_c(Bbb Z^2, text{site}),
p_c(Bbb Z^2, text{site})big )$, with probability 1 all words are seen.
We also show that for some constants $C_i > 0$ there is a
probability of at least $C_1$ that all words
of length $C_0n^2$ are seen along a path which starts at a
neighbor of the origin and is contained in the square $[-n,n]^2$.
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