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Register a new userEmbedding binary sequences into Bernoulli site percolation on $mathbb{Z}^3$
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Bernardo Nunes Borges de Lima
Publication date
2013
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English
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We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $mathbb{Z}^d$ with parameter $p$, known also as percolation of words. In 1995, I. Benjamini and H. Kesten proved that, for $d geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d geq 3$. In this paper we consider $d geq 3$ and $p in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence, for which every run of consecutive {0s} or {1s} contains at least $M$ digits, can be embedded.
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We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.
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