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

de Lima, Bernardo N. B.; Sanchis, Remy; Silva, Roger W. C.

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

Authors

de Lima, Bernardo N. B.

Sanchis, Remy

Silva, Roger W. C.

Description

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) $ξ\in \{0,1\}^{\N}$ 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.
10 pages, without figures