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

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

2009-05-28

arxiv.org/abs/0905.4615v1

Mathematics

Probability

10 pages, without figures

Scientific paper

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.

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