Percolation of words on $\Z^d$ with long range connections

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de Lima, Bernardo N. B.
Sanchis, Remy
Silva, Roger W. C.
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Abstract
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Consider an independent site percolation model on $\Z^d$, with parameter $p \in (0,1)$, where all long range connections in the axes directions are allowed. In this work we show that given any parameter $p$, there exists and integer $K(p)$ such that all binary sequences (words) $\xi \in \{0,1\}^{\N}$ can be seen simultaneously, almost surely, even if all connections whose length is bigger than $K(p)$ are suppressed. We also show some results concerning the question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related results are also obtained for the question of whether or not almost all words are seen.
Comment: 10 pages, without figures
Keywords
Mathematics - Probability, Mathematical Physics, 60K35, 82B41, 82B43
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