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

Abstract

Consider an independent site percolation model on ${\mathbb{Z}}^d$, 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) $\xi \in \{0,1{\}}^{\mathbb{N}}$ 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